Spongy brakes problem

I have bled the car brakes three times now, cannot see any air in the
system, but the brake pedal still feels a little spongy, and you have to
depress the pedal a bit further to get the brakes to work as before.

I have checked the brake pads and they all have lots of material left for
another few years before needing replacement.

The only thing new in the system after I started the brake fluid change
is a brake caliper I had to replace (for a new one), I have not checked,
but would assume it to be working as it is new and such a simple device.
There are no apparent leaks in the system, at least, none that I can
find, and none at the wheels.

What could be making the brakes spongy compared to before I started when
they were fine, but _needed_ a brake fluid change? Could the brake
caliper be stuck and cause this spongyness?

I have been using a "one-man" spare tyre pressurised brake bleeding kit,
not depressing the brake pedal until all the brakes have been bled.

Any help appreciated.