If you want to do this without having the end grain be the face of the coasters, here's a method you can use. It involves creating a template you can use with a router and flush trim bit to make copies.
First, prepare the work pieces that will become the coasters the same way you did before: using your bandsaw, cut a rough circle. Cut it a little big, maybe 1/16" - 1/8" larger than the final dimension of the coaster. You'll clean it up by tracing the template with a flush trim bit in your router.
To make the template, cut a 4" blank using a hole saw drill bit such as this one. Hole saws leave a plug of wood, but will also have a hole in the middle of the plug where the pilot bit cut through, which is why you can't just use the hole saw plug as a coaster directly. (Unless you're okay with your coasters having a hole in them! Perhaps it's a feature, not a bug!) Clean up any roughness on the edges by sanding. I would use my bench sander for this, but you can do it by hand, too. The template should be made out of something about 1/4" thick. Hardboard/Masonite is my usual go-to for this.
Next, you'll have to attach the template to the work piece. The standard way of doing this is by using double-sided carpet tape to tape the template and work piece together, but after you're done tracing the template, the carpet tape will leave some residue that you have to clean up, and more importantly you have to be careful the template doesn't move as you push against it with the router. WW.SE user Graphus suggested a better method of attaching the template. Take blue painters tape and put it on the surface of both the work piece and the template. Then superglue the two surfaces together. Unlike carpet tape, there is almost no risk of slipping, and no residue is left on the work piece after you've taken the painter's tape off.
Now adjust your router so the bearing on the flush trim bit will spin against the template. Route around the template, removing the excess.
Finally, pop off the template and remove the tape. You have a round coaster! The template can be reused to cut as many as you care to.