Surfing Waves

I'm looking for something I can wear over my fairly thin thermal rashguard to maintain core temperature on those days when it's a bit warm for the wetsuit but too cold for the rashguard.
I was thinking maybe a compression shirt or vest with fleece lining for outdoor activities.
(reasoning is, they are cheap, and machine washable)
Would that help at all?
Any other recommendations?

Wetsuits work by trapping water against your skin and letting your body warm it up, and blocking the wind.

Fleece lining works by trapping air against your body (and blocking the wind).

I think a fleece top would probably just become waterlogged. To the extent the water is kept against your body and warms up, then it will help, but if you are wearing it over the top of your rashie then I'm not sure how much it will assist. Might work a little bit but will add some weight.

Compression shirts are usually designed to wick away moisture, so would work the opposite of the wetsuit. Not sure how this will go, never tried it.

Why not just get a 1mm or 2mm top? No point in saving a small amount of money for a solution that constantly annoys you

I'd go with dtc's suggestion of the 1mm/2mm top. They don't cost too much and would be the best thing for what you are after.

the issue is the financial investment, because for 90% of the time here (US NE), a wetsuit is needed...last year I wore the rashguard only maybe 3-4 times...a thicker one I would wear maybe 8-10 times at most during a year.
Maybe a neoprene vest to wear over the rashguard? That could also be used under the wetsuit during the colder months, I'm not too happy with the padling performance I get while wearing the current rashguard under my 3:2 wetsuit .

update: found a zippered vest ridiculously cheap on ebay, will try it out over my rash guard. front zipper shouldn't damage any board, right?