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.
It can be very difficult to change hosting services. It is sad that hosting cannot help to transfer website files from one hosting to another. I tried to do it myself, but as a result, I broke something and now my site does not work. Do you know if it is possible to ask a web hosting service to help transfer all important files? Because I don't really want to figure out what my problem is. My friends said that the problem is most likely in DNS and I was even able to find a kind of very useful article about DNS https://www.softwaretesttips.com/dns-probe-finished-no-internet/ , but I very much do not want to look for the problem in DNS .
From personal experience, I can say that transferring a site to a new hosting is usually not a difficult process. Often, the host's support actively helps you with this. But for the first time after transferring the site, I advise you to enable site monitoring. This way, you can quickly track potential problems and quickly respond to them.
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.