There are many words starting with the prefix "ex-" that add an "h" (exh-) when the next letter is a vowel. The appearance of the "h" in these words seems to be pretty arbitrary at this point. But at some time there must have been a common genesis which either lost or added the "h" within these paticular words. I suspect there was some sort of inner metanalysis (see "metanalysis": http://wordsmith.org/board/showflat.pl?Cat=&Board=miscellany&Number=29152) at work here. But I'm not sure that term would apply to what's happening within a single word. And why didn't they all add/lose the "h" at the same time by the same process?

Some examples of this: exhort, exorbitant; exhilerate, exhibit, exigency, exist; exhale, exalt (many more)