Generating a session on metasploit

I see an exploit on msfconsole that matches the OS(linux) of the target. I already have target user access; I copied the ssh private key in order to gain ssh access without having to reuse the original exploit. I did NOT use msfconsole to gain user access. Now I am in the privilege escalation phase. The exploit I found explicitly offers privilege escalation to root(linux) .

My question is this:
The MS exploit requires a ‘session’. How can I add a session through an exploit that isn’t used through metasploit? How would I do this if I can ssh into the target as a non-root user?