1) Check the bypass spring cap. While this is a minor issue I have seen it on my Melling pump. The cap for the spring should be held by the cotter pin. However, I have seen it quite a bit in further from the cotter pin, under tension from the cap. This isn't a very big issue as it would just push your max oil pressure higher, which is not needed.
2) Check the pan clearance. This can starve a pump of oil f it is too high or too low. Personally, I use silly putty to check the screen depth from the bottom of the pan. you can use silly putty, play dough, a drill bit, etc. Keep in mind that if you aren't checking with a gasket installed that you have to ad the gasket thickness into your measurement. Shoot for 3/8" to 1/2" from the screen to the bottom of the pan.
3) Check your oil screen/pick up fitment I have had a problem with oil pump pickups fitting correctly. The pump to pickup fit is an interference type fit. I have had problems with the pick up a few times. I don't know if it was install error or part error, but on my last pump I went through 3 pickups before I got one that was secure. Just because you hammered it in with force doesn't mean its sealed 100%. If you can move it easily, trash it and get a new one. A lose pickup will suck up air and cause problems. After you think it is secure have it welded or brazed on. I would go with TIG, but in the absence of that a MIG would work. Brazing would work as well if you know how to do it... if not you can burn a pump. Remove the bypass spring prior to prevent the spring from weakening then reinstall it. I recommend you strip the pump down to the bare casting and the pickup for this.
4) Check your drive dowel height. I have seen where the dowel for the oil pump drive can be too high and hold the oil pump up from the block, preventing proper sealing. Check this with out a gasket. If you can pivot the pump on the block something is wrong. Try to tap the dowel in or grind it down. Keep in mind that this is best done on a block that hasn't been through the final wash yet as any metal shavings in a new engine is a bad idea.
5) Check for block to pump interference. This is something I found myself and unlike the previous checks this is something I have not found documented previously. I don't know if a certain year range blocks are effected by this or a certain oil pump. I have checked this with 4 different melling oil pumps and three blocks. Three of the oil pumps were the Melling standard volume, M81A, and one was the Melling High Volume. The blocks I checked against were two 2000 model year blocks, and one 2001 model year. All the pumps and block had the same issue which I had missed in the past. On the inner most point of the block there is a stepped machine cut. Looking from the bottom of the block the very rear side of the pump can interfere with the block, lifting the pump off at least .065". This will prevent the pump from sealing to the block creating an oil bypass which can have disastrous effects. I believe this to be the cause of my cam bearing failures.Its easy to miss. You can bolt the pump down and not notice it. The only way I noticed it is by rotating the pump and feeling the pump hang up on it. Using a feeler gauge under the pump will also show the problem. Install the bolts to align the pump without he gasket but do not fully tighten the bolts. The pump should be flush to the block. If the block hasn't been through the final wash I would grind that little bit of block down to prevent that same issue with a new pump. If the block is already clean grind the oil pump then wash it. The images below give you a better idea of the possible problem area.
Block:
Modified HV pump:
in goes the Mopar pump, fits perfectly. Gonna button her up and see if this was causing my Hot Idle OP issues( 65psi startup 40-50 psi cruise and5-10 psi hot idle) I will give you a update soon. Just curious Silver XJ, how was your OP before and after this fix?