IF - The camera; The Shooting Distance and The Focal Length of the Lens are chosen; the only way to manipulate the DoF is by changing the Aperture.
In the shot of the Woman holding the Flowers (F/5.6 @ 1/125s @ ISO200), I would have to know the Focal Length and also the exact Lens which was used, to reasonably calculate if all the Background could ever be made into acceptably sharp focus, by simply changing the aperture.
But yes, if the same shot were made at F/16 @ 1/15s @ ISO200, then more of the background would indeed be in acceptably sharp focus.
WW