My secret is paprika powder. It changes the whole flavor profile. A fair amount is needed, and it needs to be pretty fresh powder, not something that's sat in the cupboard for two years.

Alton Brown's technique [1] of using two skillets is also great if (like me) you don't have a panini press. You put the sandwich in one skillet, heated, then put the bottom of the second skillet on top.

[1] https://altonbrown.com/recipes/perfect-grilled-cheese/