If you can change the A record’s TTL value to a lesser number, it will shorten the time required to propagate. Be a good netizen and don’t leave it small forever.
Ask your old host if there is a temporary URL you can use to access the old site while the appropriate DNS is not available. You’ll need to change the WP home and site URL settings, which is elusive in multisite. Use temp-domain.com/wp-admin/options.php to get a full alphabetical options list.
Realize that URLs embedded in post content still point to the original domain. You shouldn’t need to follow those links anyway.