A fine thing to do, Gawain, and welcome -- back -- to the, ah, the melee.
Well, let's look into a few things that influence what's done -- basically, how much room you have in there to work in and to fill. A good general idea to get started on is your brow measurement at hat sweatband level, and a like circumference of the helmet, even getting the tape measure on the visor if necessary. At Lists-legal minimum for the SCA, we hope for 3-3.5" larger in the helmet, with 4-5" more perhaps more comfy. Eventually the difference becomes overly great and the helmet is becoming too large for you.
Somebody here is going to call "You've got a bascinet -- how about a period, sewn-in, bascinet liner?!" with much enthusiasm. It does have the inestimable advantage of riding cooler if of linen, and of being useful as a show-and-tell in public demos.
A period lining also neatly hides a last-ditch foam-spangen-straps resilient component -- it's too open to be called a layer -- for in depth defense. Useful against inexperienced Marshals ("Oh, there's closed cell foam in there, good!") and heavy-hitting Dukes.
The innards at least of the period liner are more shaped to your head than to your hat -- think
quilted Snoopy-hat. So is the foam-spangen, whose strips may be ductaped together into a spangen-frame and then the whole held in place in the helmet skull with more ductape before the liner gets sewn in. Perhaps you might add a block of foam to fill in at the point of the basc. Leaving plenty of room for air in there is how a period, or suspension, bascinet liner actually works, though.
To install a period liner, you need the packet of upholstery etc. needles Tandy sells, and use of a Dremel to make the 1/16" or so holes along the bottom edge of the basc and up the sides of the face to stitch the liner in using nylon leatherworkers' heavy thread for stitching awls. The thread comes in white, brown, and black. Dental floss also works, and may be colored however you like rubbing it with a crayon or with a beeswax cake. Late pigface bascs included taking the stitch holes across the top of the face also. Earlier models seem to have used three widely spaced holes, rather higher off the brow edge of the face, and apparently independent single stitches/loops knotted there and the thread ends cut off close. For the rest of the holes, at the edges, sew the liner in using a saddle stitch (two running-stitches, in opposite phase), which is very strong and has redundancy if one thread may break. The other still holds.
Searchbutton the site on "period liner" and I think you'll find pictures too. It's a hood, really. Camail liners are quite like the cowl of the hood and flow out from beneath the helmet under the camail. They need not be thickly padded, and we doubt they were. Some people may like to eyelet them for better air circulation.
The bad thing about a complete foam layer in your helmet is it's an insulator and doesn't shed the heat of fighting one whit. The sweat of your forehead cascades into your eyes instead of being absorbed and wicked away. Open-worked, strategic foam and just a bit of it doesn't give this problem.