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"Aavikolta" would be more typical in this case.
Place name inflections in the "from" sense are quite arbitrary in Finnish, I don't think there's any consistent rule for them. You can be from aavikolta and mereltä and Tampereelta, but metsästä and ilmasta and Turusta.
Finnish is a little more literal with the suffix -ssa (in) than English. Aavikossa means to be in the desert the same way as meressä means to be in the sea.
From the desert can be either aavikosta or aavikolta, but the former means to literally come from under the sands whereas the latter means coming from the desert area.
Kiitos! Great info about aavikosta vs aavikolta.
"Olen aavikolla" - "I'm at the desert"
"Olen aavikossa" - "I'm in the desert"
So, +lta is for something you're at and +sta for something you're in. For aavikko I'd always say at / +lta. I think in most cases you just have to know.