When it comes to coils, this is a tough one to answer. Certainly for different types of detecting, having a smaller or larger coil can be an advantage. But 90% of the time the stock coil will give you good results for coinshooting.
There are two main types of coils available, concentric and double-d (DD). Knowing which type your detector uses helps you understand how to get the most out of it. A concentric coil puts out a cone-shaped field (shaped like a O), the DD field is more like the "center slice" of the concentric (shaped like an I).
With the concentric coil, the deepest part of the field is very small, so if you are searching for a deep dime-sized signal, you have to really overlap your sweeps to maximize the depth. The problem with this is that the interference from shallower objects and trash will keep the detector from even "seeing" these deeper items. This is called target masking.
The DD coil is far superior in my opinion, as the narrow width of the field cuts out a good percentage of the trash and signals near the surface and the narrow field can really punch down around the trash signals. Also by scanning an area at 90 degrees and 45 degrees to the original pattern, you will be able to find almost all the targets in an area, and target masking is minimized.
A small coil is said to be good for trashy areas. With a concentric coil, this has some truth. The small coil will help you cherry pick a few good items from the trash. I think for the most part, smaller DD coils are no more effective than the stock coils. Most often though, I see people reporting making more finds (or the "look what I missed!") with a smaller coil, but it is not the coil that made the difference! What happened was they tightened up their sweep pattern and had to slow down with the smaller coil. Had they used the stock coil, overlapped their sweeps and slowed down they would get even better results.
Large coils are touted for having great depth, the trade off is they have a larger field so they are much more prone to target masking. Not only do these larger coils tend to be heavy, but they are really only effective in super clean areas due to masking. People buy these coils because they think they need more depth, and while this may be the case at certain sites that have been filled, reworked, or if your area has a high "sink" rate for items, a bigger coil is not going to help you at most sites. In most areas almost all coins are within reach of your stock coil! Any depth advantage is negated by the effects of target masking.
Where a large coil has advantages is in covering a large relatively clean area. A field, a beach... finding traces of an old homesite in the middle of woods. These areas all have low trash concentrations, so there is little target masking and you can cover more ground faster with the larger coil.
Depth is, for the most part, a bogus issue. If you are coinshooting, most coins in "natural" ground (grassy areas) are less than 12" deep. Most newer name-brand detectors made in the past 15-20 years get adequate depth with their stock coils. The reason we do not find all the coins is not because of lack of depth, it's rather from poor overlapping of sweeps, target masking, and inferior target ID meters on the machines.
Before you go out and buy a new coil, especially for a machine that is new to you, learn how to use the coil that came with the machine. Understand what coil you have and how the field of that coil works. Slow down, overlap your sweeps. Remember that the coil that comes with your machine is selected by the engineers to give the best overall performance and also provide good weight and balance for your machine. All the features of the detector are tuned to work optimally with the stock coil, not that 5" hockey puck or the 24" trash can lid!
Master the stock coil and your detector, first. Understand your site and how the coils work. If then you decide you have a site that would benefit from use of a different coil, weigh the cost of the coil to the potential finds and see if that accessory coil is really a good investment. I think most often you will find that improving your technique with the existing equipment, and digging more junk out of a trashy area will have a greater impact on your finds than a new coil.