According to this calculator at 16mm, my hyperfocal distance is at 0.7m at f/11. So on the safe side, if I wanted to make sure the scenic background is tack sharp I would focus at 1m or so. But now the question is, what if I needed things to be sharp in front of 1m? Say a flower in the foreground and a mountain in the background?
To get deeper DoF at the same focal length and focus distance, you need to narrow the aperture. Another option to get something too close in the foreground into the DoF is to move your shooting position back until the desired element in the scene is within the DoF for your selected aperture.
First, you must realize that depth of field (DoF) is always based on a particular viewing size and distance as well as focal length, aperture, and focus distance. The calculator you linked to is set up for an 8" X 10" print viewed at 10" (25cm) by a person with the manufacturers definition of viewer's eyesight. If you want to display the picture at a larger display size and viewing distance, the hyperfocal distance will change for any given aperture. If you are viewing the image at 100% on a monitor with 72 ppi, a 24MP (6000X4000) full frame image would be displayed at about 83" X 56"! At that viewing size and a 50cm viewing distance the DoF would be much less that of the 8" X 10" print, and the hyperfocal distance at f/11 for a 16mm lens would be 3 meters. For a 1.6x APS-C camera the hyperfocal distance would be 4.8m.
If you calculate your hyperfocal distance to be .7M, subjects as close as .35M should be in acceptable focus. A flower at .4M for example should be in acceptable focus. A flower at .2M likely is not in acceptable focus. If your subject is closer than .35M, then you either have to modify your aperture, use a larger format sensor, use a wider focal length, other techniques such as focus stacking to get the shot.
Part of your original assumption is incorrect I believe. If you want to make sure that the background is tack sharp by focusing at 1M instead of .7M, you are also pushing out the nearest objects that are acceptably sharp. You should be fine focusing at the hyperfocal distance, while still achieving a tack sharp background at infinity. Some people do say to focus at the hyperfocal distance, then a few feet further to "be safe". I think this is OK, but I would rather just stop down a bit if I am only at f/11 to begin with. If you are already at f/20 or similar you might not want to do that, and you probably don't need to anyways.
I think this question is already covered in: What is "Hyperfocal Distance"?