One way to do it is you can configure your site to connect to your new database, remotely; this way whether your visitors are directed to the new site or the old site, all the new updates will be saved to the new database. Now the new database will have all of your current information and can continue to update.
This technique is the most technical, but would minimize downtime as much as possible for all visitors to your site. You or your web designer would need to make changes to your site during the transfer to make this work.
A simplified set of steps include:
- Transfer all the files, databases and other settings over to the new server. (Via free content transfer or you transferring the files yourself.)
- Test the site on the new server to make sure it works as expected.
- (If changes have occurred to the database between steps 1 and 4, then re-transfer the most recent version of the database. This step is unnecessary if you do step 4 immediately after steps 1 & 2.)
- Change the database settings on your site so that the site on the old server accesses the database on the new server.
- Change the name servers and DNS to point to the new server.
Using this technique, it would not matter which server a visitor sees while the DNS propagates, since both the old server and new server update the same database.