Ok, before I respond to this I want to fully understand it because I'm not sure I do so I'll rephrase:
The US rules for saluting the flag state that if you are in full uniform you would not remove your hat to salute the flag. However, if the hat you are wearing is not your uniform hat you would be required to remove it. The only official hat is the following one:

The above being a correct interpretation of what you have stated, I must disagree on several counts on the third sentence in that statement. One, the BSA sells several different uniform hats and they don't state that the other hats may not be worn as part of the official uniform. The don't make that statement for a very good reason -- their sales would drop
Two, it is customary throughout history for uniformed organizations to create distinctions amongst different sections of those organizations by adopting hats and other smaller uniform pieces that are different from other sections. Amongst Boy Scouts, neckerchiefs also serve this function.
Three, any hat specifically issued for or mandated for use by an official program of the BSA is an official hat. The most common example of this would be summer camp staff hats but there are many others. I have at ten or twelve of these types of hats from various official functions.
Fourth, based on substantial historical prescedent, I believe any troop has the right to choose their own official hat (and neckerchief). While that right could be taken away if the BSA decided different, I doubt they would. Or at least I doubt they could get away with it because of the hue and cry it would generate amongst the membership.
Now, if a Scout was wearing a Mack Truck hat or his dad's old hunting hat, yes, he should take it off, but if he's wearing a hat that has an acknowledged role within the Scouting program, then he should salute with his hat on.
Does the BSA even sell official shoes anymore?