the logo on the right is their old logo, and the one on the right is the new one

Mmm.

sorry the one on the left is the new one and the one on the right is the old one.

I like your take.
Three notes:
1. Capital S is a little weak next to the other letters.
2. Original one still has a better rhythm to it.
3. Your logo could be seen as more (or not enough) of a departure from the old one than the client could be confortable with.

thanks for your feedback, I think all valid points.

Any suggestions on how to make the s stronger and improve the flow of the logo as a whole?

I agree with the capital s.

I don't agree with riccard0 about the rhythm. I think your piece conveys a kind of calmness that I suspect is suitable for the client.

Great job!

One problem I see about the S is that the thin lines appear thinner than the ones of the lowercase.