Hello Parasso,
What I would recommend is the following:
1. Choose a new host and setup your account.
2. Mirror your site with your new hosting, while maintaining your existing host.
3. Once the site is setup and tested at the new host, then change the DNS at your registrar to point towards the new host.
4. Allow 48 to 72 hours for the DNS to propagate before turning off your old hosting.
This should deliver a pretty seamless move for you.
Good luck,
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