I can only speak from my own experience (limited) from
working with Waltham pocket watches.
My size 18 vintage 1888 needed a mainspring.
I found one on ebay for about 20 bux.
These springs fit into a “drum” and the outside of the spring
has a small “T”. The inside has a small square hole.
Be careful about which way, clock-wise or Vise-versa it goes.
Attached is a pic of my 1888 lever set Waltham.
Bob, the master, is correct (always) in that a fine file and steady hands
can renew the catch on the barrel hub so the spring doesn’t slip
off. It’s canti-levered a wee bit so that the spring holds tightly
to the hub.
I hope the pic helps all out there who may have similar issues.
