I have a refrigerator in my garage that has been plugged into a wall socket with an
Extension cord for several years. It draws 12 amps. The only circuit breaker that shuts off this socket is a 60 amp main.

I want to move the fridge to the opposite wall but the socket there is controlled by a 15 amp breaker that has at least 3 amps of lights already so I am pushing the limit.

Is my best bet to have an electrician put another socket drawing from the one from the 60 amp socket on the other wall?

Nope
Theis refrigerator needs to be (should be) on 20 amp circuit... no more, no less. And your 60 amp service just isn't adequate for today's electrical needs. You need a major upgrade.

The current (no pun intended) minimum service is 100 amps for a residence.

However, if you are going incur the cost of upgrading to 100 amp, you might as well upgrade to 200 amp and be done with it.

By the Way, this is a residence, isn't it? Its not mobile home is it? If it is a mobile home, then different rules apply.

Mobile homes usually have 200 amp service in them.

Mobile homes usually have 200 amp service in them.

All of them?

I didn't see where he said his house was supplied by a 60 amp main. Possibly a separate garage with 60 amp subpanle or something.

fridgq, clarify the status of this 60 amp main please.