From the Spanish "xaquima" to the AmE "hackamore"

The earliest occurrence in print of hackamore I have found, in this cowboy story from 1850 (very likely the source alluded to by Wikipedia and Etymonline), explicitly associates it with the Spanish term. ('Pete' is 'Dutch' or German, the 'old man' is apparently Mexican.)

“When a broncho is lassed, he is fust choked down, then a hackamore is put on him. Know what a hackamore is, Pete?”
   ”No, I tidn’t.”
   ”Wall, a hackamore is a Spanish halter, that is made so as to slip when a rider pulls on it, and draws a hoss’ nostrils together and shets his wind off.”
   ”It’s a jáquima,” said the old man; “an invention of the Moorish Arabs.”

Hackamore = /hækəmor/ is a quite natural and until you get to the very end almost inevitable Americanization:

• The stress pattern of the two terms is identical: 3-1-2, where 3 is primary stress, 1 is wholly unstressed, and 2 is secondary stress. This significantly conditions the treatment of the vowel in each syllable.

• As sumelic points out, /h/ is the English phoneme most similar to the sound represented in Spanish by ‹j› or ‹x›. Just for instance: the Cardinals baseball player Julian Javier was referred to by his manager and teammates as "Hoolie".

• Spanish ‹qu› is pronounced /k/, which is exactly reflected in the conventional English spelling ‹ck›.

• The stress pattern of the two terms is identical: 3-1-2, where 3 is primary stress, 1 is wholly unstressed, and 2 is secondary stress. This significantly conditions the treatment of the vowel in each syllable.

• Spanish ‹a› is in fact pronounced /a/—more or less like US Southern "I" or the so-called "broad a" used by older speakers in Boston—not as many people think /ɑ/ ("ah"). This is a sound between the General American phonemes /ɑ/ and /æ/ (as in "cat"); since stressed /ɑ/ is fairly uncommon in US Englishes (except as a realization of "short o"), it tends to be realized in Americanizations as /æ/—compare "Laramie", "Alamo", "Los Angeles".

• The Spanish /i/ is tenser and higher than English "short i" (/ɪ/), and that's sometimes realized in English as /i/—in fact, Google Books gives us an 1876 instance of a spelling with ‹y› which represents this:

  “… they had found us, me an’ Arturo, lyin’ still in the rode, me holdin’ the hackymore, which wur broken”

  But it's just as likely for this very-low-stress vowel to be completely reduced to the schwa, /ə/, which is what the spelling ‹a› represents. Even with an ‹i› spelling we tend to reduce this way: inhabitants of Mexico, Missouri, for instance, say /'mɛksəko^u/.

• The only piece left is ‹a›→‹ore›. Open /a/ and /æ/ do not occur word-finally in English, and even /ɑ/ is rare except in foreign-language names. In Americanizations it usually becomes /ɔː/, with or without a corresponding change to the spelling; consider "Choctaw", "Kennesaw", "Waukesha", "Wichita". And in fact there is at least one instance of a spelling with ‹aw›:

  Hackamaw is a sort of halter, or headstall, made of the end of a lariat rope and put on in such a manner that it holds the head of the mustang firmly without the danger of choking the animal. —Dennis Collins, The Indians' Last Fight, n.d.; largely memories and accounts of the 1870s, but published in 1913 or later.

  I can offer no example of /ɔː/→/or/. Ilmari Karonen's explanation, that ‹ore› is a conventional spelling for /ɔː/ in non-rhotic dialects, suggests one possible influence; another is that /or/ is a very common terminator in English in words with this stress pattern. It's also worth considering that English does have a handful of naturalized names and words ending in /əmor/—paramour, sycamore, Baltimore, McLemore—and of course the Hickamore-Hackamore riddle, which seems to have been first published by Halliwell-Phillipps in the 1840s.