Thanks, JungleBoy!
There is the alternative of the train (CP). The local train stops in Barquinha, then Tancos, then I think there is a stop right AT Almourol, and then Praia de Ribatejo, which is the most easterly town in the Barquinha municipality. The new trail is probably quite close to the rail line most of the way along. Certainly it is at Tancos.
So in theory, one could take the train one way, and then walk back.
Looking at the map, it looks like it's only about 4 km of the entire 10 km trail as far as Almourol--about halfway along the trail.
And looking at the schedule, you're only travelling 2 stops for the 4 km; you can buy the ticket on the train, which will only be a euro or two, and they run in both directions several times a day.
On the schedule below, you're looking at the "Regional" lines, at small stops between the major ones of Entroncamento and Abrantes.
Heading east (to walk back west), the current schedule seems to show Barquinha pickups at 7:52, 11:33, 17:33 and 19:38 (pretty late). The run between stations literally takes 5 minutes.
If you wanted to walk, and then ride back, the pickups at Almourol are 12:05,15:54, and 20:08.
Remember, these are slowish "milk-run" trains without reserved seats. They're meant for locals, and quite used to picking up passengers without advance notice.
Bom caminho, and welcome to Vila Nova da Barquinha and the heart of the Ribatejo!
PS--Map: