I go out there 3 or 4 times a year for work. And have for a decade or so.
There’s decent riding in phoenix itself, certainly for a half day at a time, well signposted trails, probably 6 or 7 spots with a variety of tech. Hawes mountain is my favourite but south mountain has more tech. Browns is pretty flat, more gravel than xc.
Other than that Sedona, Prescott and tuscon all have good riding, the latter you can get a service to take you to the top of a decent of over a mile in height (lots of pedalling still needed!)
Weather in Oct is still warm, but normally bearable. Probably peaking on a hot day around 100f so worth getting your rides in early/late.
Grand canyon is day trippable, if you’re fit you can get to the bottom and back in one go but it’s a long way (26 miles, 1700m vert from memory). It’s a well signed trail, and not technical, but is fairly risky to do solo, I think I saw maybe 5 other people significantly far in. If you go and don’t want that risk, do hike a couple of miles down the trails, the top is really busy but (no exaggeration),95% of folks don’t leave the very top, so the atmosphere 10 mins down the trail is completely different. There’s also good, technical hiking in phoenix, camelback is a good one to test your legs, bit of a scramble in parts, all well signed posted again
Oh and regardless of what you choose, you want a hire car. You can’t get anywhere in Arizona without one, and whilst Uber is pretty good, your own car is way way easier.