We rented the old residence when we bought our current house. The carrying cost was low, and rental depreciation was a factor. Following may not apply, generating passive income held a greater value to us than the rate of return from maintaining the property as a rental.
With the usual disclaimer – not an advice – in case you decide to rent your old house… and if the numbers work one can consider pulling equivalent of the ‘tax-free’ money from the old house to purchase a second rental (house/condo?).
Regarding the rental process it is typical to have 30-45 day turn over between tenants, unless the prospective tenant is from out of state and can move immediately. If the cash-flow supports, renting $300-$500 below market also helps in keeping the tenants for > 1 year.