PID controller for a very wide temperature range (∆T=190°C)
Generally speaking, the PID should be able to handle that requirement without having to break it down into subranges.
It can be tricky to find the right values for a PID.
Check the answer of this post about the process of adjusting the PID, it's easier to go step by step rather than randomly changing the values.
More detail can be found here.