What watch was it?
Modern quarts timed electric watches, the
second hand is times to complete one revolution per minute. It moves in a continiouse arc around the dial, not stoping at any one number.
Old clock-work watches have an escape mechanism; the
second hand moves in a series of jerks, between each number increment, halting briefly between each latch of the escape mechanism, timed by the pendulum.
On cheap clockwork watches of antiquity, the escape mechanism, would latch every second, halting hand by each number. More expensive watches to give a smoother action, would increment in maybe 1/2 second latches, while Rolex were very proud that their perpetual, self winding watches has a secondhand that moved in 1/3 second increments.
I have a Jeager, and people keep telling me its stopped, becouse the 'second hand' goes round the dial once a month! (the outer dial is the date hand; seconds are measured on a small inset dial)

Anyhow.... after all that effort to make a mechanical watch second hand move smothly, like it wasn't being latched, being perfectly achieved by quartz timed electronic watches..... people grumbled that the perfectly smooth continiouse arc was a tell of a cheap watch... and wanted ones that had second hands that moved in jerks again! (good going Rolex!)
So, watch makers started using stepper motors, that did just that and halted the second hand briefly between increments.
So, the starting point is, does your watch actually stop every second or half second? Becouse if not.... the 60 pictures will be the only way to get the effect you want, as other wise the hand will merely define a blur accross the dial.
Next; may be easier to remove battery, or let spring wind down so that the watch is stopped, and move it on 1s a time to take each picture, rather the trying to catch 60 individual frames at exactly 1s intervals as the needle goes round! Read the instructions.... it may not even be possible..... or easily.
I seem to recall an ad photographer, explaining a prize shot for one of the watch makers explaining they had to remove the watch glass in order to photrograph it without reflections, and to be able to move the hands where wanted.
But, I have taken a 60 second sweep by long exposure of a small electric alarm clock.
Annoyingly my timing was off, and I never managed exactly 60 second... I got just under a full revolution, or just over a full revolution, never exactly a full revolution... and I got blur between the main increments, and a double ring, as the tail of the second hand was counterbalanced.... and I seem to recall removing the glass against reflections.
However, did 'work' give or take, the stopped hands, remaining over the numbers long enough to record an image.
Helped that the second hand was white, and the face black; exposing to get the second hand, I think I got an EV for a 1s exposure of the whole clock; estimated that the white hand would probably still record at 1/4 second, so stopped and filtered down to 1/4 of the original 1s EV, then held it on B the full 60 ish second.... I think. It was probably near twenty years ago I did it... and I cant find scans of the pics..
But those are the principles, and first thing is to know what the clock or watch is doing, mechanics wise, and whether the second hand will even stop to record an exposure by each number for you.