You should go to a good TKD school (hiding the fact you are muay thai) we have tons of stretching techniques.
Dynamic stretching i think is the one you are looking for. For example i remember kicking to where it was confotable to me 70% and then kicking higer, and higher trying to hit a target. I thought it was just trying to kick harder but it was part of dynamic stretching.
Also you can do many exercises outside, many stretching, but the real thing is to practice the kicks for along time every day.
Some tips that helped me:
1) stretch 3 times a day (morning, afternoon, night)
2) PNF and Dynamic, so kick higher for dynamic (feel the stretch)
3)You can open your legs (kind of like a split) all that you can, then hands on the floor change them into your waist and lift your back stright.
4)jump with knees up, jump on one leg attemting keeing with it.
5)Run, sprint, run, sprint, run etc
6) legs = hands. We did this on TKD like 4 months before one of my tournaments Mexico Vs Korea i think, And our job was to think that our legs = hands. So we turn light switches off with them, close the frige or doors, play soccer (very good for legs if you know how to play), we started kicking in the middle of nowhere (like while we re cooking) and we practice 3 times a day.
7) Practice, practice practice the kikcs.
Now, all of this help me. The stretching was hard both pnf and dynamic i remember i cried a few times of the pain while i was doing it and i was unsure if the coaches knew wtf they were doing but they are coache for a reason and i trusted them. Hey i got 1st place and got to go to the centroamericans games...anyways i guess hard work pays off so you can take our advizes but only on training your ass off on what you are weak (kicking higher) you will see diferences.