Godaddy will hold the domain after is expires for a period of several months. If your renew domain more than 19 days after expiry, then there is a redemption fee added. This redemption fee can be sizable. Godaddy's help pages say that the latest you can recover the expired domain is 42 days.
Even after the 42 days, the domain may not go back onto the market. I have seen the register hold domains for up to 6 months before they could be re-registered.
When the domain does come on the market, every domain I have ever watched has got bought instantly by a domain hording company that puts spammy adverts on it.
I would think the idea of allowing a domain to expire and then register it again elsewhere is a very dangerous task. The domain will be out of service for about 6 months and there is a very high risk that someone else will grab the domain before you when it does become available. Better to transfer in the normal way.