Don´t know which engine you have, but both 4.0 and 4.7 have a MAP sensor. On the 4.0 its mounted on the throttle body and on the 4.7 on the front of the intake plenum. MAP is the number one input for fuel injector pulse. I don´t know of any cleaning of this sensor and the fact that the engine works fine when cold, i won´t expect it to be dirty. It could be faulty though.
But without observing the live data, it will be a shot in the dark.
If you have an android phone, you could download the free app Torque and just buy a cheap bluetooth dongle for the wireless connection to the jeep. This app can show you MAP, O2 sensor(s), inlet temperature and coolant temperature and many other live data, which could add in the trouble shooting.