Though my '95 DX has the remote lever and fold-down-rear-seat features, I have noticed on occasion my key doesn't work in the trunk. #1: that can vary by key: older key can be trickier to get working.
Aside from my older key being a challenge sometimes, I notice that sometimes a key requires reversing (remove, rotate 180 degrees, and re-insert) in order to work better, or I have to angle the key up or down to get it to work: it will seem to be blocked from rotating until I do so, then open with a bit of effort as if forcing it just a little. Play with it.
Recommendation: call or visit a vehicle locksmith service to help ensure your key works properly, perhaps to get you a good key made for the trunk. They may tell you the lock mechanism is aging and needs replacement. Personally found a local mobile guy for those needs ... and back in the day when most people used Yellow Pages, his ad was small, not flashy, as were his prices but his service was great. Cheap unless he needed to make a new key by probing the lock.
Note some cars actually have keys that work in doors and ignition but not the trunk: a "valet" key to keep guest drivers out of your trunk; I have no idea whether Corollas ever had those.