The last rifle I bought from the CMP was just a service grade Springfield but it came with new wood, too. Gonna try and use it for a 3-gun match somewhere this year.
Here's my Bavarian rural police CMP M1 carbine in action at a 3-gun last year.
If you are looking for a club that is affiliated with the CMP, I highly recommend the Garand Collectors Association (http://www.thegca.org). For a measly $25 yearly membership fee, you get a great quarterly magazine too.