I stumbled upon this n-gram recently in the course of my work. For those who don’t want to click, “outa” was the dominant spelling of this slang term until the 1960s, when usage of “outta” rose up. They were used in about equal measure in 1980, at which point “outta” shot up to become more than 3 times more common.
What happened? Why do people prefer “outta” now?
I dunno. It’s how I’ve always seen it, not having learned to read until 1959. Don’t recall seeing “outa” much.
Ive also never seen outa, only outta, if someone isn’t spelling out “out of”, which I see the most.
I wouldn’t draw any conclusions from a pair of percentages. Could be a typo for some other word that appeared just a few times. Hard to say without context and absolute numbers.
I’m pretty sure I’ve seen “outasight” or “outasite” written more frequently than “outtasight” or “outtasite”. In fact, if I search for “outtasight” in Google Books, it suggests “outasight” instead.
Things change.
Maybe because to a naive eye, outa looks like a foreign word (Greek perhaps?)
I remember reading books when I was a kid written in English, but where the author had dropped markers in conversations to indicate that a speaker was French. Thus, a character in direct speech would often start his sentence with “Oui”. I didn’t know this word, and so for years I wandered about mystified, thinking that French people said “Oy” for no apparent reason a lot. (D’oh).
Looking at the actual page results, “outta” was much rarer than “outa” for most of the century, with hits mostly being OCR misreads of “Gutta” as in the rubber-like substance “Gutta Percha.”
The big change might be the release of NWA’s Straight Outta Compton in 1989.
I started reading in the early 1960s and I have no recollection of ever reading “outa” prior to this thread. So I question the OP’s thesis at least a bit. ETA: Exapno’s research results weren’t there when I posted. That changes things a bit.
Assuming arguendo the OP’s thesis is valid …
Perhaps it’s by parallel formation with gotta? “Gotta” as a contraction (correct term is ???) for “got to” makes some obvious sense.
“Lotta” for “lot of” or “outta” for “out of” makes less obvious sense alone, but makes some sense as a parallel. If English wasn’t so averse to ending words with “ah”, I’d expect “outah” and “lottah” or “outuh” and “lottuh” to be the preferred formations.
I too never saw “Outa” used till this thread, only Outta.
In fact, there’s a storage facility on Long Island NY that I have driven by from time to time, called “Outta Space” (maybe its a chain, there’s one in Australia too) 
When looking that up just now, I found there’s a 1971 song by Billy Preston called “Outa-Space”, so clearly people did use that spelling at some time (there’s also a 1999 song by that name, indicating some people still hadn’t learned how to spell it right even by the end of the 20th Century… 