First, even a proper CLA won't necessary make all speeds exact. I've seen, multiple times, test results after CLAs where tested speed might be off by as much as 30-50% on the slow end. And I'm talking about test results provided by good repair techs. I assume that the mechanics of making one speed correct might alter a different speed. And I assume it also depends on both the initial build quality and the subsequent wear on the shutter.
So being a 1/3 stop off at 1s isn't has much a concern (at least if it was my lens) as a 1/2 stop off at 1/60.
I'm no expert, but it does seem a CLA is in order. But, out of curiosity, did you discover the discrepancy on film or just with your tester?