I have very fine hair as well.
I sewed a hair comb into the seam of my mantilla (see picture below). I pull the hair at my crown together and hold it with bobby pins (I usually do this in the car. A barrette would work as well) and then I slide the comb in. It told very well since the comb in held up by the excess hair and the bobby pins.
I used the same approach with my daughter’s chapel cap, but I sewed a small clip barrette to the seam instead of a comb. I have used her cap a few times, and this holds very well, too.
If this does not work, a thin headband sewn to the veil would work as well.