Altium - How to Convey Pin-In-Paste Technology in Design
No, Altium doesn't have a feature for this. As for the amount: You need to specify it just as you said by adding a feature (e.g. a Polygon or a rectangle) onto the relevant paste layer. You might want to do this in your PCB symbol.

Altium doesn't make any assumptions on the thickness of your stencil (especially not if it's a stencil with multiple thicknesses) so it cannot decide the amount at all, that's up to you.