Disbelieving the illusion doesn't make it go away.
Figments and phantasms, when disbelieved, become transparent with a faint outline. Disguise self (which the Hat of Disguise uses) creates a glamer, which actually alters your appearance. If someone makes the Will save against the illusion, the illusion is still there; they've just become convinced that it's unreal. Recasting the spell to create the same illusion again will change nothing.
How is that going to convince anyone?
I'm Lord Ulrich the Tall and you're a guard at my tower. I walk up to the gate, you salute and lower your spear and say "Hail Lord Ulrich", and as I walk past, I forget to duck under the doorframe. Instead of banging my head against the doorframe, the top two inches of my head somehow goes right through it. You roll your Will save and conclude that this is not right. "Halt!" you say, raising your spear.
So I fiddle with my helm for a second and you watch in horror as I change, my bone structure and body contours shifting. I lose about four inches of height and forty pounds. I turn around to face you and I'm now Lady Agatha, Lord Ulrich's wife of ten years.
"Oh," you say, "well, that explains it. Hail Lady Agatha!"
No, what you do is raise the alarm! This is clearly some shapeshifting demon or sorcerer that needs to be killed with fire. And send someone to check on Lord Ulrich, because he's probably dead or tied up in a basement somewhere.
Suppose I'm wearing a mundane disguise, like a fake nose. We're talking and I sneeze really hard and my fake nose falls off. So I pick it up and put it back on. Do you now think it's my real nose?
Once they know you're wearing a false appearance, any reasonable person is going to be suspicious of both your appearance and your motives.