Time Traveller's Pocket Watch

After getting one from the quest and using it up, how do I go about getting a second one?

the monsters that drop andys watch drop the TT watch instead

They just drop from monsters in Leafre.
I got one from Beetles and one from Dual Beetles.

Beetles seem to have best drop rate. Dual Beetles drop rate is a bit less. you will need about 8~10 watches total