The majority of beach or water hunting questions I receive are always search coil related, especially from people wanting an edge at heavily hunted beaches.
So my answer may surprise many, but the search coil that came with your metal detector is that advantage they are looking for.
In my opinion, the search coil that came with your metal detector is always the best option for general beach or water hunting.
I like to think the people who designed and built the metal detector, chose the search coil size because it has the best sensitivity and depth to a wide range of targets.
To me, larger and smaller search coils for my metal detectors will always be accessories for special beach or water hunting situations.
I know many beach and water hunters prefer only using extra large search coils on their metal detectors, but I would rather go smaller than larger in more beach and water hunting situations.
In fact I class a 10 or 11 inch search coil as a large search coil, because I am so fond of using small search coils.
Some of my favorite beach and water hunting sites are difficult to detect using 10 and 11 inch search coils because of iron and target masking, change to a smaller search coil and I am constantly digging targets.
I have many sites I am always busy digging targets using a 6 or 8 inch search coil, and I see people using larger 10 and 11 inch search coils walk right over those same areas without stopping to dig targets.
In my beach and water hunting related books and these blogs, I talk about the importance of sweeping your search coil low and slow.
The larger the search coil you use, the more important your sweep speed becomes, especially if you are using 10 to 12 inch size search coils.
A slow and low moving search coil helps to cut down on both target and iron masking, helps with improving target IDs on detectors with screens and increases your chances of detecting deep targets.
Of course it should go without saying, that you should first learn how to use your metal detector with the search coil that came with it, before attaching and swinging a bigger or smaller search coil.