Reciprocity laws seem to have come a long way but I don't think there is a multiple state permit unless you're a federal Leo or something.
In Texas it's a simple process. Basically if you can legally own a handgun you can get a license. A few additional rules apply like no dead beat dads ect. Being able to decipher some simple but vague instructions helps. Enough disposable income to pay for the classes, firearm, fingerprints and license fees is needed too. The funny thing is, law enforcement professionals I've spoken with know very little about chl laws and stuff, instructors ain't much help and the info they have is usually out dated. The best thing to do is to pay attention each new legislative session for the new laws. The standing laws can be found easily online.
There are many myths, misconceptions, half truths and misinformation out there. This information is easy enough to find, but you have to dig some.
The statutes on arming yourself spans several codes. Penal codes for one example and goes as far as the labor codes.
Hand gun laws are updated on a regular basis now; and most are focused on restoring our rights after years of knee-jerk legislation.
It's your right and it's your responsibility to study the rules. Not a cops, not an instructor and definitely not a guy online.....