Hi, David,
With the high speed train now stopping in Segovia, that's what I'd do. I think at least one a day goes to Leon from Madrid with a stop in Segovia.
As far as pilgrim friendly hostel, I have stayed in many in Leon and one of my favorites is the Pension Blanca. Breakfast is included in the price, and you also have free use of the kitchen. It's about equidistant from the train station, the cathedral, and the parador (where the Salvador breaks off from the Frances). But I do think its prices have risen lately, so you might just want to check booking.com for specials.
When I walked the Camino de Madrid (which goes through Segovia), I stayed in the Hostal Don Jaime. Very close to the aqueduct and the old part of town.
http://www.hostaldonjaime.com
Lucky you, walking the Salvador, you will love it! Will you continue onto the Primitivo? Buen camino, Laurie