Traffic has to be routable from one network to the other. This means that there must be some device with both a 10.x.x.x and a 192.168.x.x. address. This device must have routing enabled between the networks to pass traffic between the networks. Traffic flow will be bidirectional since SMTP uses TCP and there are acknowledgement and control messages sent back and forth between the hosts involved in the transaction. If the server in question has both addresses on a single NIC, the SMTP service must be configured to listen on the same network as the sending SMTP host. You can test this by running a telnet session to port 25 on that host to see if the SMTP service answers.
–Reply from LoneWolfBW
The issue turned out not to be that (per my post, I had the SMTP relay installed on a server with two NICs, one on each network).
I had set up the SMTP relay to forward to the IP address of our SMTP server. That was what was failing per below.
I changed the setting to the FQDN of the SMTP server and that works.