Closed the closet door and Kaboom the acorn thumb latch broke off. Looking for a solution to open it
A screwdriver (suitably small/thin)
A wire
An ice-pick (right, nobody but old folks has those)
A nail
Pretty much anything that fits through the slot and allows you to lever up the latch bar. That will be easier if you can hold the door in such a way that there is little to no pressure on the latch.
Point the tool of whatever sort downwards as you push it in, then tip and lift it upwards.
It appears that even if you were to remove the hinge pins, the fit of the door in the frame is such that you probably can't pull that side of the door outwards, due to interference by the barrels of the fixed hinge.
If the latch broke, you may be able to just pry up the remaining bit of the latch, if that is still in place.
Depending how the doorframe on the latch side is configured, an old credit card inserted between the door and frame below the latch and moved upwards may also work.