You can increase yield per square foot exponentially by growing vertically. This might include growing a pallet garden, on a trellis or netting, hanging baskets, hanging grow bags, or GardenSoxx hung sideways on some shelving. Really, it's up to your imagination.
Pallet gardening is really unique. Here's a link to learn about it. Basically it is where you take a pallet, staple some landscaping fabric around part of it, and throw in some plants and dirt. Maybe it's not that simple but take a look at the link above, it's pretty cool.
GardenSoxx(TM) is a sack of soil. You cut open a couple of holes and plants something. It can't get any easier than that. I saw a commercial strawberry grower using these six high which increased his yield per square foot dramatically higher.
Hanging baskets are old school but you can hang them almost on top of each other is you have the right configuration. Something like this link.
Hanging grow bags are pre-built bags of soil, like this one, where you cut it in small slices and add a plant.
Remember, only your imagination stops you. That's how all the above methods started.
Enjoy your garden!