I found a simple fix for this issue on my phone. If you have the same settings as I do, it might work for you, too.
Settings—->Passwords & Accounts
—->Choose the offending (Hotmail/Outlook/Live, etc) account
—>Account (email address)—>under Outgoing Mail Server, click on SMTP. If your settings look like mine below, I clicked ‘OFF’ the Primary Server smtp-mail.outlook.com....leaving smtp.live.com ‘ON’.
I don’t know why this worked, as I didn’t change the actual offending Incoming Mail Server, but my phone fetched my mail just fine after I did this. Maybe it’s because the Incoming and Outgoing server names no longer match?
I hope this helps someone else. I really didn’t want to go through the whole ‘delete account’ process!