JLCPCB, PCBway, NextPCB, ; all of these are capable of manufacturing more than 10-layer PCB. So, 6-layer is not a big deal for any of these manufacturers. Yeah, one thing I should take into account considering your location that I can see in your profile. If you are looking for a manufacturer in Europe or UK in specific, none of their shipping charges might be economical to you. In that case, you can consider Eurocircuits. They manufacture upto 16- layer. So, 6-layer is well within their capacity. They have an office in the UK. Contact all prospects and select the one that you think will be the best suited for you.