If you just want to use Disney transportation then you have a few options.
Bus from DTD to FW but then you need to take the internal FW buses to the trail ride. Never had luck with either of those being quick.
The route I always use and some may disagree is we take the Magic Kingdom bus from SSR then from the MK we take the boat over to FW. It seems super fast in the mornings because everyone is going opposite the direction we are going. Usually takes less than 30min this way. Not sure if its best later in the day. We try and do TE for breakfast at least twice each trip and this route has been the most successful.
We are
DVC members and our home is SSR but I grew up at the Fort so I get back there as often as possible. Give it a try and if you find something quicker, post it and let us know.