Cox Email using Outlook when in Mexico didn't work

I use MS Outlook as the client for my cox email account. Recently when in Mexico I received email fine but any email I created in Outlook and sent was never received by the recipient and I never received any error or undeliverable message. I would create the email offline then connect to an un-secure Wi-Fi network at a roadside restaurant to send the emails. Outlook said it sent them, they show in my sent folder but no one ever received them. Any thoughts? Thanks Chuck