The best way is to add a completely new subnet and move devices in a controlled manner. It gives time to test and resolve problems for the challenging devices - even leaving some as they are if they cannot be changed.
But you might also consider other criteria for example: fastest, cheapest, cost effective with some risk etc.
Your idea of expanding the current to a /22 can work but the potential risks/complexities are:
the use of 192.168.0. and 192.168.1 should generically be avoided to clash with other default installations on routers etc.
make sure you new range is compatible with VPNs and external parties you connect with.
Keep the default gateway device in the 192.168.1.x range so that the current devices can communicatee with it before their subnet mask is changed. This also allows a device to not change their mask if you cannot access it.
You must use a router/firewall as your default gateway that allows hairpin (in and out same interface) - so that during migration devices with the /24 mask can communicate with devices on new /22 mask. This is the most challenging bit.