Ideally, you'll want to avoid these contracts until you get the High-Altitude Flight tech node and the improved jet engine that comes with it. But since when do Kerbals wait for ideal conditions? Here's my example of dealing with the situation.
I started by attaching two LV-909 rocket engines to a regular low-atmosphere plane:
In this save, I don't have custom action groups yet, so I'm using the RCS control to toggle between jet and rocket modes. I also set it to enable the cockpit's reaction wheels to improve control at high altitude, and close air intakes to reduce drag. Only the fuel tanks attached to rockets have any oxidiser in them. Strictly speaking this optimisation is unnecessary, but it can win you a bit of range.
Approximately 12km up the main engines stop providing enough thrust to maintain the climb, so it's time to switch to rocket mode. In my case, it brought me to 20km up at a 45-degree angle, but it is probably not optimal. It may take some practice to hit your target area this way, because you'll be picking up additional horizontal speed, and thus it's possible to under or overshoot.
At this point the plane plummets, but as it goes deeper into the atmosphere, it regains pitch authority, and at 10km it's enough to regain control and turn it around for a safe landing.
Of course, this thing has very limited range, speed and acceleration because of the extra weight. Another problem is jet engines stealing fuel from the rockets' tanks - you may want to pump the fuel manually (requires an R&D centre upgrade) or add some pipes between them. Or try using SRBs instead.