Warm up and some mobility exercises make you be mentally ready for the training. So I often add some mobility exercises at the beginning of the personal training session. You may want to try some from the next session.
Hip Flexor Mobilization
Left / Right knee in front of the body, making sure both legs are straight to start with. Tense your glute muscle, bracing the core muscles, then finally move forward. The goal is for hip mobility therefore, you don’t want to let the lumbar spine start moving. Count for 2 seconds and move back to the original position. Do this about 10 reps and the other side as well.
Hip Flexor Mobilization with a bench
This exercise is very similar to the previous one. But you are targeting Rectus femoris muscle more since the rectus femoris muscle crosses two joints. To do this, the starting position is the same as the last one, then grab one of your feet, then move forward. Making sure the movement is from only the hips but not the spine. That means movement is very little.
Overhead Lunge walk
Arms up straight, bracing the core first, stabilizing the lumbar spine throughout the movement. Step forward, starting the lunge movement. Not only does this increase the stability of the lumbar spine but also this extends the upper back.
Some of t-spine mobility / stretch exercises here