As, I live in Whaley Bridge, I know the area quite well and have done lots of walks in the area. The best way from New Mills to Chapel would be over Ollersett and down into Buxworth. The best way up to Ollersett is either via The Grouse pub on Low Leighton road or the road going up near the fire station.
Once at Buxworth, the Navigation pub is worth a visit as mentioned on previous feed back. I would then go down the old tramway next to the pub, which joins a country lane into Whitehough, where you'll find another pub called The Oddfellows arms, which has great beer, pleasant landlord and a charming king charles spaniel!
Once replenished turn left out of the pub, up a hill into Chapel.
It's a great walk especially if you like BEER!!