Another alternative, which I've used in the past, is to take the public footpath near The Rambler that heads west to Barber Booth and then head south on the road to join the Chapel Gate path. This takes you to the western end of Rushup Edge which has some great views (including of Mam Tor itself as you approach) and is easy walking. You can head back down to Edale from Hollins Cross.
Lazar's suggestion of using Hope station is also a good one. When I used to travel out from Manchester, I'd often use a combination of stations on that line to do a linear walk - and the Great Ridge is a fine walk.