The 05-07 HPOPs are very reliable. Don't change it out.
There is a lot to learn about troubleshooting a 6.0L! If you start guessing, it gets expensive.
Starting better when warm is more of a FICM
(or injector) issue than an issue with the high pressure oil.
Get the ForScan app for a smartphone or a laptop. It requires an ELM327 adapter or a USB to OBDII cable for the laptop.
First thing to check is for codes. ForScan is a great code reader.
Also, you need to install a fuel pressure sensor and gauge ..... or at least have it checked. Low fuel pressure can seriously damage injectors and then they can trash the motor.
The type of coolant you run can cause expensive issues. You should first take your ForScan app and determine if your oil cooler is starting to plug up. You run the truck until it is COMPLETELY up to temperature (at least 15 minutes) and then pull the ACTUAL coolant and oil temperatures.
If the oil is more than 10 degrees hotter than the coolant, then the oil cooler is plugging up. It could fail and cause more issues.
Running hot, and running with bad injectors, are two of the more expensive issues that you can have. If you monitor the truck and pay attention to things, you can avoid big problems. If you think you can just drive it and not watch things, it will bite you and you will soon be posting that it is a POS.