If it does follow, that would indicate that you have something in the rim or the tire that is causing the flat. A thorn in the tire, for example, may not be easy to spot, but can flat the second tube in relatively short order. A good idea to run your fingers carefully over the inside of the tire to inspect for any little sharp things. Something caused that flat. Maybe it fell out when you changed it, but it could also be lurking there ready to vanquish your next tube.
Did all that best as I could. Even took the tire completely off and turned it inside out so I could have a good look. Couldn't find a thing. The leak in the tube was on the side that gets close to the road (as opposed to the side lying against the rim), and I figured out right about where the corresponding spot on the tire was, but nothing to be seen.
We'll see. Have had three rides now on the new tube, today's with 6 or 7 miles of dirt, and so far so good.
Then you're probably fine.