I can´t move my sites to a new server

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Hello all you Linux/Apache-gurus out there. I need your help!
I have my concrete5-site on a older server that is about to get shut down, and due to that I need to move the site to a newer version. (I don´t know the version of the old server, but the new server is PHP Version 5.5.17-2.)

Now, here´s my issue: the new server doesn´t support Concrete5-versions lower than 5.5.2. (I´ve installed a new version on that server and that works, but I can´t move my sites who has a lower version to the new server...)

The old server doesn´t support versions newer than 5.4.2.2, so I managed to upgrade my site (who is on the old server) to 5.4.2.2 but when I upgrade to 5.5.0 the site crasches. I´ve tried to move the site to the new server and then upgrade, but it doesn´t work.

Any suggestion on how to proceed? (Anything except rebuilding the webpage will be ok....)
Thanks!

Hiawatha73