There's not a lot to go on here, but I'll chime in anyway.
The reason it doesn't leak when it is cold, is because there is no pressure in the system when it is cold. So no surprise there. And it may not drip when hot, because it evaporates off before the drip forms.
I suggest you get yourself a cooling system pressure test kit. It allows you to pump up the pressure and check for leaks while the engine is cold and not leaking.
Then find the leak. Use a mirror if you have to, and get underneath and look up with a flashlight. Whatever it takes to see where it is leaking from.
Don't assume that a new pump shouldn't leak. Especially if it is a rebuilt. The quality of stuff we get these days just plain sucks.
Using the pressure test kit, you will know you are done when the system holds pressure for 20 minutes.
Dodge did not build my Jeep.
1988 Wrangler,4.2L I-6, Howell EFI JP-1,CRT HEI Ign,AX15 ext slave conversion, Zone 2.5" lift + 1/2" booms, BFG 31 x 10.5, Bestop.