The Constitution’s Full Faith and Credit clause should allow any concealed license holder to use that permit in all 50 states, following that state’s laws and restrictions.
However, Congress has not determined that concealed weapon licenses are worthy of Full Faith and Credit yet. Congress has done so for Driver’s Licenses, Marriage Licenses (between a man and a woman per the Defense of Marriage Act,) Birth Certificates, etc.
There have been bills proposed to add cpl licenses to the list, but have not passed yet.
And you have to beware of that because if they legislate on the issue, it would end up being a de facto federal license. With “harmonized” rules across all the states.