When a hat has a ribbon headband these are normally not robust enough to support a rolled up piece of paper, like a trilby above. However you can still make the hat smaller with some basic needlework:
1. Put the hat on, and feel where it is too big by checking with your finger at the front and side to see where the largest gaps are
2. Once you've noted where it needs adjusting, you can simply make a small fold in the ribbon and stitch the fold with a few holding stitches using needle and thread.
3. Sometimes its easier and more comfortable to make a series of small folds and stitch them rather than trying to one large fold.
4. You may want to carefully pin the tuck first before you actually stitch it and then try your hat on (without pricking yourself ideally) so you know its the right size before you change it permanently.
5. One tip, don't sew a tuck at the front of the headband (where it would touch your forehead) because the folded ribbon may be inclined to leave a little mark on your head when you wear it, which may attract strange looks whenever you take off your hat in polite company...