Does Six Sigma more than enough for a service industry to meet the client's SLA?

Officially may be as far as SLA is concerned.

However, improvement is still there even through Six Sigma, since Six Sigma is only attempting to figure out how far we away from perfection so that nearing zero is workable. If you are migrated from Poor to Fair means you are still accepting 25% flaw. If you are aiming at achieving DELIGHT, then there is a word called "Wow" and this is beyond Excellence or Six Sigma.