I can print a 4x6, then scan it and save at the higher resolution.
Why would you do that! Where is the magic process that creates new detail there? You are only losing information!
You are cheating on yourself, you probably are seeing more contrast, because you are loosing middle tones, you probably are watching the paper texture, but you are NOT getting "crisper look".
This is important because as you are confused about what you see you can not replicate it with the proper steps in post processing. As I say, more contrast, different saturation, some noise, etc.
Use L. Forget about s and m.
Apply sharpen on any program. Yes you could shoot at RAW etc. But sharpen is something that can be applied into any JPG image and makes it... sharper.
About the resolution. With your 10 Megapixels (3872 x 2592 pixels) you can print a 19.3x12.9 inches image. (a good standard is to use 200 ppi) or a 25.8x17.2 with 150 ppi. which is still good resolution.
If your image is not sharp enough you probably have a focus problem, motion blur, dirty lens, or simply your lens is not sharp enough.
You need to review your process. Probably use a tripod, focus on live view, reduce the aperture, etc, to make the image as good as your camera and lens can.
Then go to post processing and improve it.
But please, do not ever again print it and scan it.