The cheapest 'zoom' you have... is your feet....
Also, technically, what you are after is a telephoto-lens, not necessarily a 'zoom'... could be a zoom-telephoto, but all depends.
lenses come in 'standard' - those with a field of view aprox that of the attension area of the human eye; about 40 degrees or so. Anything with a wider angle of view.... is a wide angle; anything with a narrower angle of view is a telephoto.
A lens 'focal' length gives an indication of it's angle of view, but doesn't define it. The angle of view is defined by the triangle, from the edges of the film or sensor to the focal-point.
So, the 'standard' lens on a 35mm or Full frame camera, is 50mm, the triangle, 35mm base, by 50mm high giving sides at approx 40 degrees. On an APS or Crop-Sensor camera, the base of the triangle is just 24mm wide, so a focal length of just 30mm or so is needed to get a similar triangle; while on a medium format camera with 60mm wide film, the similar triangle calls for a focal length of about 80mm.
Hence.... a 50mm lens could be a 'wide angle' on a medium format camera, a standard lens on a 35mm camera and a telephoto on a crop-sensor digital camera.
Ie: a 'long' lens is reletive to the width of the camera its mounted on!
Meanwhile... back to zooms.
A zoom lens is merely a lens with a variable focal length. ie its not fixed like a 'prime' lens.
So, you might have a wide angle 'zoom' that offers a range of focal lengths and field of view in the wide angle range; or one that offers a range of focal lengths, and fields of view in the telephoto range.
Or as the 18-55 zoom on your 3100, a zoom that is a wide angle at one end and a telephoto at the other, and a standard lens in the middle.
Right; moving on from the semantics of definitions.... back to opening statement; cheapest zoom you got is your feet.
A longer, higher magnification lens does not get you 'closer' to your subject.
Subject distance is whatever it is, between where you plant your plates, and whatever you want snap of. Longer lens just makes the subject bigger in the frame.
Getting 'closer' to your subject does the same thing.
And here's the thing..... if you want to take pictures of wild-life... birds and stuff... a subject on which I have to confess I am no expert.... the 'skill' as I understand it is in getting as CLOSE to that subject as you can.
Standing back with a big lens? Not really encouraging you to get up close and intimate, is it? Meanwhile; not encouraging you to get up close... standing back... more room between you and the subject... and lets get more specific here... birds. Birds live in trees. Trees have branches. More space between you and bird, more trees, more branches, more 'guff' to get in the way and compromise your composition.
Meanwhile; technically; the apature f-number for a lens is calculated from a ratio of the hole diameter in relation to the lens length. Lets say you have an aperture, a hole, say 18mm in diameter; on a 70mm lens, that might be f-4, but on a 210mm lens, an f-5.6.... same size hole, same actual aperture diameter, but the f-number changes as the lens gets longer, as the narrower angle of view is letting less light through it. This is why zoom lenses often have two f-stop ratings, for their fastest aperture; ie a 70 to 210mm lens rated as f4/5.6. The iris has a max aperture diameter, and doesn't care what focal length is set.
Now, as you are likely to have noticed, 'fast' lenses, ones with big apertures or low f-number ratings tend to be expensive. Wide angle lenses, can have fairly fast apertures, quite easily; because to get a low f-number the hole diameter only has to be a big proportion of the focal length. So if you have a lens that's say 25mm in diameter, you can quite easily get a 20mm hole in the aperture iris and an f-number down in the 1.something range, if the focal length is around 20-30mm. If you want a 'fast' aperture on a 500mm lens... you would need to have an aperture size something like 200mm to get an f-number around 2.something..... That probably means having a chunk of glass on the front of your camera 250mm or nearly a foot in diameter to get an iris in it that can open up that far! Big BIG lenses. And usually very expensive ones!
So, as you venture into telephoto lenses, aperture sizes tend to get more restrictive, and start having higher f-numbers, while prices start going up.
Smaller apertures, will restrict the amount of light getting into the camera; hence, for any ambient light level or exposure value; you will have to use lower shutter speeds or higher ISO sensitivities.
But... at the same time; longer the lens you use, more magnification it has on the subject; so it will also magnify any motion blur. Small angle of camera shake hand holding a camera with a 55mm lens is likely to not move the subject more than a pixel or so in the frame of the camera. Hand Hold a 500mm lens, and same angle of shake could move the subject completely OUT of the frame! So.... you generally need to increase your shutter speed to avoid motion blur, either camera shake, you holding the camera, or the subject moving during exposure. Rule of thumb, minimum shutter speed is your lens length; So, if you are on 50mm of your kit zoom, you can probably go down to a shutter speed of about 1/50'th or a bit less. Go to a 300mm lens, and you probably don't want to use a shutetr speed under 1/300th, possibly any lower than 1/500th, as longer lens is likely to be heavier and less well balenced and less easy to hold as steady.
So, your longer lens is likely to limit you from using faster apertures or slower shutter speeds, and push you towards using higher ISO numbers to compensate... BUT THAT usually means lower defanition......
Always a trade off. and the further you go, the less helpful it's likely to be.
Back to first principles; you want to get 'closer' to your subject, your first 'zoom' and your cheapest zoom..... is your feet!
And if you have been at the game barely a month.... liklihood is that you are far from an expert stalker..... or an expert photographer, and there is almost certainly an awful lot you can do to get better pictures with what you have, if you think about how best to use it and work on aquiring skills not kit.
However.... the 55mm on a crop is not particularly telephoto, and doesn't offer much reach. And birds are inconveniently small subjects...
I couldn't have got much closer to that little Sparrow... like I said, I am no expert on bird photography! I was snapping my kids in the park in the snow, and the thing came to see if we wanted to feed it, landing maybe 8 ft away on a tree over hanging the fence behind me and the kids pointed and yelled to get a picture..... so I did... and that's a crop from the middle, so enlarged about 8x!
This suggests a few things; 1/ On such a small subject, the camera hadn't got a clue what to focus on. 2/ Even at pretty close range, you obviously need a fairly long zoom to get much frame filling.
So, obviouse starting point would be the Nikon 55-300 zoom, that's about £300 retail, though you do find them discount for about the £200 mark. f/4.5-5.6, its 'only' about one stop slower than the kit 18-55; which means you might only have to go up one stop on ISO for the slower aperture, though at the long end; 300mm rather than 55, means a shutter probably of 1/320th instead of 1/45th, so up three stops on ISO to compensate for shutter, and you are looking at using ISO6400 a lot, rather than ISO400....
And you's STILL need to use your feet to get closer and get better shots without trees and stuff in the way; and patience and stalking skills to get your shot.
But it would be a start I suppose.