When I try to send email from a particular mail account on my iPhone, I get a message that says: "Cannot Send Mail / The connection to the outgoing server "smtp.amjdigital.com" failed. Additional Outgoing Mail Servers can be configured in Settings>Mail, Contacts, Calendars."
I tried re-establishing the mail account on the iPhone; but still can't send email.
I can send email using the account from another wifi source, though.
Search for the topic "Alternate SMTP Port" -- your configuration is likely trying to use port 25 to send outbound messages which is blocked on the Verizon network (there's a whole discussion on why this is the case and why it is a sound security policy, so I won't go into the details here).
You need to find out from your mail provider (amjdigital.com) if they have an "Alternate SMTP Port" for sending mail (this is sometimes port 587, but I've seen others as well -- and I did a quick check and the standard ports for the alternate don't seem to be active on that server). If they look at you cross-eyed, ask them to have their security person take a look at why companies should be configuring an "Alternate SMTP Port" for sending mail. It's pretty simple to do. Once you have that information, you'll need to go into the advanced settings for the mail account and change the sending port number for SMTP outgoing mail to the port number they provide.
This change will work whether you are on the Verizon network or elsewhere.
Within the past 3 months it changed at Verizon.
Really recently? Perhaps it was configured to use an alternate port and for some reason the remote location turned off the port unknowingly (maybe a software upgrade or firewall change) or you reconfigured your mail account on your iPhone?