Oh, am not concerned ...
So the subject line of this thread has morphed ... no longer a 'managed VPS hosting provider' ... but an un-managed ... where you, the OP does it all.
Hmmm ... we've come full circle me thinks.
"When I'm happy that the new site is working, I'll migrate my website to the new server without the Moodle 2.6 version."
Do you mean you'll change DNS to point to new hosting ... the 3.8.1+? (which by that time might already be behind a version)
Uhhhh ... if you do ... the 3.8.1+ site would have been developed with what FQDN and cert? While being developed moodle will use the config.php variable for $CFG->wwwroot. All internal links would have used that variable .... thus in DB. So, you'll have run search and replace to change all those.
If you follow your own history ... 2.6 -> 3.8 ... you'll wait another 12 versions before moving away from 3.8. Wonder what Moodle will look like by then?
Strongly suggest breaking that history ... do initial install of the 3.8.1+ using git - since you are now gonna manage it.
https://docs.moodle.org/38/en/Git_for_Administrators
My 2 pennies!
'SoS', Ken