I am looking at Bicycle Caps. I would like recommendations as to the style that would be most comfortable.
My head is shaved and my bike helmet is not soft enough.
The benefits of a shaved head are quite numerous :-)
Here is a (cropped) picture of my helmet.
If you need a peak/brim/visor, then a traditional cycling cap is really the best option. They generally have one panel of cloth that covers the scalp backward to the elastic at the rear. The parts above your ears are their own semicircular panel. These are also known as casquettes (from french) and not "caskets" which are to be avoided even though they sound similar.
Baseball caps are to be avoided - they tend to use 6 triangular panels sewn to a single point on the top, which is closed by a snap or permanent dome/rivet. These are uncomfortable and dangerous in a collision, and can interfere with MIPS sublayers functioning.
Finally, if you just want padding then look for a skull cap or a buff. Skull caps are close-fitting and may have a brow line that follows a normal hairline while dipping down to cover the ears.
A buff is essentially a shapeless tube of light/thin material with some elastic properties. Both should sit under a helmet without adding pressure spots.
Do be aware that your helmet will need to adjust outward to allow for the extra material between you and the sub-frame. Sweating will increase because ventilation decreases.
A final option is to stop shaving and allow some natural hair to grow back - this may not be a feasible solution for all readers.
The rear/sides of your neck are also exposed - some people like to use a "legionnaire" style of sun-shade around the back. This one is a cycling cap not a baseball cap.
For sun protection of your head under your helmet during the summer, many brands make exactly the product you are looking for: You should search for terms like Cycling Skull Cap or Bike Summer Skullcap or Cycling Summer Beanie.
Make sure to explicitly check the sun protection qualities of the material used (usually included in the description).
I cannot post links to specific products because we don't generally answer shopping questions.
For riders with very short hair a triangle folded bandana tied at the back of the head works well to prevent sunburn through the helmet vents and serves the other purposes of keeping a layer of sweat next to the skin for cooling and reduces the amount of sweat that falls into your eyes.
A bandana is tiny and can be stashed in a jersey pocket in case of an unexpected sunny day.
Cotton/poly-cotton isn’t an ideal sun-screen but works well for this particular use. Probably darker colors are better.
I'd rather stay away from the entire light mounting system safety discussion, but one solution may indeed be to replace your helmet. You can fix the comfort problem without having to get an additional cycling cap. Your existing helmet looks quite old, and the padding is likely wearing out. The foam gets saturated with dried sweat and starts losing its springiness, and the fabrics can start getting scratchy. Also, the shape of a helmet arguably has the biggest impact on how comfortable it is. Finding one that better matches your head shape means there's fewer pressure points and you're less reliant on the padding to provide you with comfort.
If possible, going up in price range can help as well. Here's a photo of the inside of my mid-range helmet (Giro Source MIPS). There's visibly more and thicker padding than yours, and the fabric has smaller pores, which will feel nicer against bare skin.
Additionally, if you insist on mounting stuff to the front of your helmet, getting a helmet which reaches further down the back of your head can help stabilize the whole assembly. It's difficult to stabilize an asymmetrically-loaded hemispherical helmet without cranking up the retention strap tension or the chin strap tension, neither of which are going to help your comfort issues.
Sunburn is best dealt with using a very high protection sun block. If you head is getting burnt though the helmet vents, you should be applying it to you face and neck anyway.
Marino wool skull cap is my personal choice for an under helmet covering. These come in very thin single panel and will provide similar feel as hair between helmet and skin. There are 'high tech' synthetics that work as well as Wool. Avoid anything with thick seams, and anything too thick (unless you need if for cold).
