In order to make the Solaris server relay messages to another host involves editing the /etc/mail/sendmail.cf file and setting the value:
# "Smart" relay host (may be null)
Obviously replace "mailserver.my.domain" with the FQDN of your real mail server that you want to send email through. Restart sendmail by running:
svcadm restart /network/smtp
This setting will allow mail that originates on "sol10" to be sent out, but does not help when you want other devices on your network to use sol10 as it's relay. The answer was surprisingly easy:
Create a new file /etc/mail/relay-domains. In this file, put the networks you want sol10 to accept email from. For example, if you have devices on the 10.0.0.0/8, 172.16.0.0/16 and 192.168.20.0/24 networks and want to use sol10 as the relay, enter the following lines in /etc/mail/relay-domains:
Once done, restart sendmail again (same command as above), configure your clients to use the Solaris server as their SMTP server and check the output in /var/log/syslog while you send a test message.