SSL Multisite Login Problem: Subsites Just reload login page

I've recently switched my Wordpress multisite over to https and it's now causing a problem with login.

When I try to login to any subsite, the login page loads OK, but when entering the correct credentials, it simply redirects back to the login page. No matter what I do, I can't get past the login page for any subsite.

I can login to the main site through the root login page, and then navigate to the sub-site dashboards, but it's very frustrating for users that they can't go to their own sites directly.

I'd appreciate any help anyone can offer on what I've done wrong!