The name of hosting provider actually doesn't matter much since the problem of how to transfer web hosting from one company to another has a common solution. Thransfering FROM GoDaddy is almost the same as transfer website to godaddy which you probably already did in the past.
Anyway, the actions you need to perform is:
1) move your site's files to a new hosting
2) move the database to a new hosting (it can be easily done through web interface which must be in any hosting service nowadays)
3) redirect the domain DNS to a new hosting
Still, the DNS change can take up to 48 hours, so there is no guarrantee that your site won't get offline for some time if you've less than 2 days remaining on GoDaddy.
You can try this. 1. Login to your client area of GoDaddy and choose the transfer domain button 2. Then click on the domain transfer button. On next page fill the fields like new registrar, transfer date etc and submit. 3. You will receive an email from GoDaddy. 4. The next thing you need to do is contact your new registrar and transfer the domain. You can use the transfer key method. It is faster than EPP. Hope it helps.