The YMCA on Hillsborough Street is the bomb. They tore down the old building and built a new one that's been completely finished for almost a year now. It has all the usual suspects + a pool, racketball, basketball, spinning, a barber, massages, aerobics, steam, sauna, etc. I know that they are open early. The monthly fee for a single member is reasonable.
YMCA-Alexander Family YMCA-Membership
I live part time in Glenwood South and it's walkable by passing through a section of the Cameron Park neighborhood. It's not as close as Peak but it's much better. It will take you less than 10 minutes to walk there depending on where in Glenwood South you live.
I have been a member there since 1991 and highly recommend it.