I would say if the hot tap complies with the details in API-653, there's no reason to modify it later.
If it doesn't comply with the details in API-653 or API-650, there may be no simple way to fix it.
But note that API-650 does allow for some variation in design.

I always considered hot taps to be temporary and that they should be replaced with API 650 nozzles at the next opportunity. If the tank is out of service, my choice would be to bring it into compliance with API 650, especially if it is connected to hard piping. My opinion only...

Thanks IFR, and yes, the tank will be out of service. There is external piping attached to the nozzle in question.

When you say "replace with API 650 nozzle" are you suggesting that we should cut out the existing hot tap nozzle and install a new nozzle? Is there a way that we can trim and weld from the inside to fully support the existing nozzle?

Agreed with JStephen, the nozzle can be permanent if the hot tap connection was done per API-653. If not, it should be replaced during the tank shutdown to bring it up to the Code.
Comparing the current hot tap nozzle with the API Codes to see if the construction is complied.

mk3223
The nozzle hot tap was completed per API-653 based on records kept at the time.

By their nature, hot tap nozzles may not be as strong as regular API 650 nozzles. Yes it is per code but why risk it? Some owners and jurisdictions specifically require hot taps to be replaced at the next outage. Especially when live piping is attached and high shell stresses may exist. There must be a reason API 650 does not use the API 653 hot tap weld details. API 653 does not use API 650 details because it can not and a 3-way tie in to the surface of the shell plate just does not seem to me to be as good as a full pen weld. Personally, I'd replace the nozzle neck using API weld and nde details and put an API repad on it. API 653 is silent on the service a hot tap nozzle can be used for, it only provides guidance on the installation of a hot tap so it is up to you and the judge you are in front of to interpret or decide what was intended. Perhaps a 6" nozzle is small enough, the actual weld is high quality everywhere, there are no notches to increase stress, corrosion is not an issue, thermal and other piping stresses are low, your tank is not in critical service, the tank will not settle, the shell plate has no laminations or other internal defects, the full pen weld to the repad is an equal replacement, etc etc, and your risk is small. I'm conservative, perhaps too much so, but your tank is a very large capital investment meant to last for years under obvious and unforeseen conditions. You have little legal risk because you did it per code but again why chance it? How much will it really cost considering the big picture and how easy will you sleep at night?