Some random thoughts:
Liners (and comparable extra socks in shoes) only work if there is room for them. If they result in making either your gloves or shoes tight, then the restriction in the blood supply will only make things worse. Loose gloves are much warmer
Silk was the wonder material up until the eighties, but polypro is much better. 'Terranova Thinnies' are pretty much standard amongst climbers.
The gloves must be absolutely windproof when it's cold. Your apparent wind speed (bike speed plus headwind) will always be more than someone just walking.
If you have flat bars and thumb shifters (or grip shifts) try wearing mitts, which are always far warmer than any glove. Alternatively if you need more finger dexterity for Ergolevers etc, there are some two finger 'lobster claw' gloves that are almost as good.
Finally, you will never get warm hands by treating the hands in isolation. Cold hands are merely a symptom that the body is cold and is restricting its blood to the core trunk / brain areas. Preventing heat loss in other areas (head / neck / thighs / front of body is the most important way of keeping your hands warm! Again windproof is the most important feature as moving air strips away heat far faster that convection or radiation (blow on hot food). But to stop getting 'sweaty' (fast way to get really cold) look for clothing with the wind proofing only on the front, or made of very breathable (but windproof) materials like Pertex, plus have ways of regulating your temperature by removing items easily (hats / buffs /zipp off arms) or opening zips and fasteners.
(Materials like Goretex have traded a degree of breathability for water proofness and while they are the best in the rain, you will still over heat in them. Other waterproof / breathables are even less good. Venting is still really important)
If you still have frozen hands after all this (including a good body base layer), you can get heat packs (little sachets) which you start by 'cracking' them and these can be slipped between you inner and outer glove across the back of your hand. They are recharged by boiling when you get home. I've found these a help on really cold days in the mountains.