I think that if my food budget were $10.50 per week, I would probably do a pantry building process. Week one would be oil, vinegar, beans and rice; week two might include miso paste, mustard seeds (you can make your own mustard and the seeds are cheaper), beans, rice; week three might be beans, rice, root veggies. week four would probably be a repeat of week three, and then week five would be looking to replenish any 'pantry items.'
Foraging is really great for herbs. Here in NZ, we have several native herbs that are good to use, as well as now-wild european herbs. Parsley and fennel grow like weeds here -- very easy to get (and you can pretty much eat the whole fennel plant). Other things are more seasonal, but can be found throughout most of the year.
Also, since the beignning, NZ towns have planted edibles in common areas. Persimmon, quince, and lemon trees are most common. Rosemary, lavender (which you can cook with), and sage are also common our green spaces. Many of our parks are fenced with berries (blue berry hedges or raspberry canes) -- and all of this is free for the taking in our community.
Community gardens are also an option for gettign "free" produce. It's easy enough to get on a working bee, and in several months (or days in our case), you'll get an email about what needs to be harvested and "take what you want." It's a great way to get simple, seasonal produce. Asparagus is on the rise here (it's spring). It's 3.25 a bunch (1/3 kilo), but it's free for us.
I wouldn't necessarily go to a mission to get food -- as I believe that is for the actual poor, not the exercising poor, but I feel it is ok to contact them and ask if there is any produce on it's last legs that they are looking to give away before it goes to waste. Waste not, want not as the say.
Finally, dumpster diving. Always sounds terrible, but you can ask your local grocer if you can have their stock that they are removing for compost. our community garden does this, and most of us simply take the best produce home, leaving only the stuff that is truly inedible for the compost bin. It's a nice way to get fruit, honestly.