If
company.org is currently forwarded to
company.com, you wouldn't want to change
company.org to the new server. It would be confusing to users who use
company.org and expect to end up at the current (old) website.
In such a case, an alternative would be to create a hostname like
new.company.com and point that to the new hosting service. Once, the new site is ready, then the existing records for the old site could all be pointed to the new server.