That depends on what you call 'old town' exactly.
Actually, Kazimierz and Podgorze lie south from that shape you're pointing at.
By Old Town I mean exactly the area which originated in the Middle Ages (during the main settlement era, mostly in result of implementation of Magdeburg Rights)https://polishforums.com/history/poles-owe-germans-53954/#msg1174498
Kazimierz is OK, yes, came into being in 14 century, has its own market square etc.
But other districts you mention aren`t old enough to me.
Some places are only Old Towns with marginal development beyond the defensive walls area:
Szydłów - once thriving, then declined and remained so till today.
Not far from Szydłów there is Staszów. I used to teach there nearly two decades ago. Great times. :)
Kalisz again. I can`t resist, it is so beautiful from the air.....
Though a closer look shows standard stuff on the ground level