Use 'an' whenever you meet a letter 'a,i,u,e,o' in front of a word.
Ex, An apple.
Use 'a' for the others.
Ex, A ship.
This is a phonetic rule, not a spelling rule, one that I think MsNyee has known implicitly since she (<--I'm guessing) began to talk.
As I said, the "standard" English is "a higher", but, based on the Google test, "an higher" appears to be valid and accepted in (parts of) the USA. I did not know that. I'm not saying I'd never seen "an higher" written before, but I had never really noticed it, and up here in Canada, it just doesn't come up. It's always "a higher".