Call on the day it arrives
Arizona requires notice before a carrier walks away, which gives you a working window rather than a crisis. Owners who start immediately place cleanly; owners who wait until the final fortnight take whatever is available. The difference in outcome is almost entirely about when the call was made.
Find out the actual reason
Non-renewal usually has a specific cause: roof age, claims history, a vacancy, an inspection finding, or the carrier leaving a class of business entirely. The last of those says nothing about your property. The others are frequently fixable, and fixing them changes what the next carrier sees.
A lapse is far worse than a higher premium
Allowing cover to expire leaves an uninsured rental and a gap on the record that follows the property. If the only option for a period is a surplus lines policy at a higher price, that is a better position than a gap. Continuity is worth paying for.