Need a process to get current clients to give us personal introductions to others
You're on the right track.
This is one of the most powerful kinds of referrals (there are four kinds, and the first to--which are the most common--suck).
I should charge you for this, but:
"Can you think back to the situation you were in that was the reason you hired us?"
"Can you remember what made you choose us give you the solution?"
"Who can you think of who is in this same situation, and would value the reason you chose to work with us?"
(There's the key question.)
"Would it be okay if you called them now to explain that, and see if they're interested in setting up a time for the three of us to talk?"
This three-person conversation is very effective.