Creating a new URL like "old.example.com" has absolutely no value here. You'll just confuse Google and run the risk of appearing spammy - they don't like "gateway" domains and it might appear like that's what you're setting up.
It sounds like you are doing 2 migrations: changing hosts, and also adding SSL. If that is correct, I'd suggest a two-step migration to minimize downtime.
Step 1: switch to the new host. Don't change to https/SSL yet - just change your nameservers to point to the new host. Depending on your registrar's settings, this is often completed in less than a day, but it may take up to about 72 hours. It's often best to do this when you have less traffic - so if you're B2B or for some other reason most of your visits happen during the workweek, make this change on a Friday afternoon and by Monday when everyone is back in the office this step should be complete.
Step 2: install SSL and use .htaccess to force all URLs to https. It only takes a line or two of code. While you are doing this step, if you do not already have redirects in place to force a subdomain - as in, if you can reach your site both from http://example.com and http://www.example.com - you should add another redirect to force all URLs to either use or not use www. That way with this step you will end up with 1 URL for every page that is enforced, consolidating all of your link juice. In this step, you won't have to wait for nameservers to propagate, you'll just want your web team to have the redirects ready so as soon as they install the SSL certificate they can pop in the new .htaccess file and it's done. (Then make sure to check your site, either using an Incognito window or a http header checking tool, to make sure everything works as expected.)
Optional step 3: The 301 redirects will preserve your linkjuice, but you can realize some small benefits from reaching out to larger sites that send you a lot of referral traffic and asking them to update their links. In some industries it's not possible to have many people go back and update your links; in others, such as when you're listed in multiple directories that send a lot of visitors, site owners are happy to keep their links up to date. This won't impact much other than giving you a very, very tiny boost in speed - visitors will go directly to the right URL instead of first trying to hit the http version and then redirecting - but if you have a few sites update your links it can be beneficial, mostly for site speed.