I have the SMTP server going "offline" problem also - on 3 of my 4 smtp accounts - actually, 3 email accounts, one with two totally separate smtp services and two of the accounts use the same Microsoft smtp.office365.com service. What happens is that all smtp works for awhile then 3 of the 4 quit with offline ssl may not be supported errors in connection doctor. The one that stays going is an old non-microsoft server kept going when the university I work for switched to office365 hosting for email. Two of the three that quit are the office365.com. One is at fatcow.com.
Looking at the Mail smtp logs. They show good connection when streams security level is none. Then fail when the security level is TLSv1. If I reset my internet gateway router, which is an old Linksys WRT54G, the smtp services all come back up. The ones that go out usually do so after a few hours of working. I do know the Linksys has an expired certificate, but I have set it to always trust via keychain manager. The cert could be a problem, but I do not think the router itself is.
I have deleted and rebuilt the accounts in Mail preferences and accounts more than once. I now have all the "automatically detect settings" turned off for all accounts as a last ditch effort. Luckily, the one old server/service stays going and will accept sends from enough of my email addresses that I am not dead in the water.
I did not notice this until I had done the 11.1 update, but did not run for long after my original El Capitan upgrade. I would guess the problem is the ssl negotiating process is breaking with newer smtp services. Hopefully, Apple will fix this quickly.