When did Jesus earn the H in the swear, "Jesus H. Christ"?

The earliest incidences of "Jesus H. Christ" that I can find occur in the late 19th century in the following story, attributed in one source to "A Texas paper" and in another more specifically to "Waxahatchie (Tex.) Mirror".

The names Jesus and Christ sound very sacred to English speaking people but among the Spanish are very common names--given and surnames. At Laredo the other day Jesus H Christ registered at one of the hotels. We remember noting a few years ago that a Mexican named Jesus Christ had been hanged for horse stealing.

This is, incidentally, not a true statement about Spanish-language naming customs. "Jesús" is a common first name but I've never heard of a "Cristo" or "Jesucristo".

Anyway, it's pretty clearly not related to the modern expression. Google ngrams has it picking up after WWI and really hitting its stride in the 30s. The earliest example I can find that is clearly referring to the deity is this (rather poorly constructed) limerick published in an anthology in 1927:

There was a young girl from Lancaster
Who'd do anything anyone asked her.
     But when she got spliced
     She got so high-priced
Only Jesus H. Christ and John Jacob Astor.

Since that anthology doesn't clam originality, we can probably assume at least some people were using the initial by 1927. Incidentally, spliced in this context probably means "married", according to Green's Dictionary.

...which leads me to the realization that I should have checked Green in the first place. He traces the exclamation to "late 19C" (I don't have the paid membership so I can't see his sources). Green does not give his own explanation, but quotes Michael Quinion here:

There have been various theories, but the one that seems most plausible is that it comes from the Greek monogram for Jesus, IHS or IHC. This is formed from the first two letters plus the last letter of His name in Greek (the letters iota, eta, and sigma; in the second instance, the C is a Byzantine Greek form of sigma). The H is actually the capital letter form of eta, but churchgoers who were unfamiliar with Greek took it to be a Latin H.

That is, IHC actually stands for three of the letters in Ἰησοῦς, but was reinterpreted as the savior's initials.

Quinion also quotes Mark Twain, who said in a 1910 autobiography that the expression had been around for a while in 1850. But we have to keep in mind that (a) Mark Twain was far more interested in entertaining his audiences than in accurately representing his childhood, and (b) people are really bad at remembering when slang came in, and tend to retroactively insert not-yet-existing words into old conversations in their memories.

So the best I can say is that the phrase was attested as an exclamation in 1910, being used as a name for Jesus by 1927, and had reached some amount of frequency by 1940. It probably came from a misinterpretation of "IHC", but that's 100% guesswork.