I know increasing the size of the photo (S,M,L) does not affect the dpi, but refers only to the pixels. My camera shoots at 300 by default, but sometimes I want a crisper look when I get larger prints. I can print a 4x6, then scan it and save at the higher resolution, but that is SO time consuming. Does anyone have a solution to changing the camera setting? It might not even be possible, or maybe I need a newer, updated camera. (I use a Nikon D40x, with a AF-S 70-300mm lens for most pictures.)
You got it the wrong way round; the dpi setting stored in the file is nowadays just a historic and mostly useless number, and you could simply change it with a tool, but there is not even any need for printing.
What you need to care for is the total resolution of the photo, which is limited by the camera hardware. You can reduce resolution, but never increase it (you can fake it by doubling the number of pixels and interpolating the missing values with noise - which is what you describe by scanning it back in). You should always use the L setting to get the maximum number of pixel the camera can produce
The resolution is defined by the camera hardware - professional grade cameras typically have 20+ Mpixel, some up to 50 MPixel; consumer cameras between 5 Mpixel and 20 Mpixel. If you want a higher resolution, the only way is to buy a better camera.
If you want to print larger, you need more pixel = more resolution. For a 10x15 inch print in good quality, you need 1500 x 2250 pixel (which is about 3 MPixel), that should be easy to achieve with any camera nowadays.
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.