I'd agree with the above that it's just more work that's making you tight. The thing is, all this work is done within a relatively narrow range of motion for the muscles you feel getting tight. So they get tight. Static stretching helps some, but dynamic stretching helps more. My solution to this, because I have the same problem whether on tandem or single, is to add Straight Leg Deadlifts to my weight schedule. I use relatively light weight, about 50 lbs., and do 30 reps. You weigh a lot more than I and can use more weight. Low weight and high reps avoids problems with the back, but more particularly with knee joints and meniscii.
http://www.exrx.net/WeightExercises/...gDeadlift.html

Keep the barbell close to the shins as shown in the gif. Move smoothly. One could also use dumbells.

These will probably make your glutes and hams hurt more to start with, but that's how it always is. I also do leg sled and squats for glutes. Another thing that works the glutes more than one might expect is pedaling 120-140 on the trainer or rollers without bouncing, though that doesn't extend range of motion.

Even though this might slow you down a little to start with, you might also try spinning faster when you climb, around 80, at least until you run out of gears.