So far the advice here has been pretty bad and does not reflect what has happened in those stores that succeeded when giant supermarkets open up next to them.
@Ally Maxwell more or less nailed it with one word - though I would stretch that to most of what has been written.
What actually works in the real world of real shops is to do a careful analysis of what it is that the large chain is NOT selling - and to ask the question why they do not sell those things.
That is EXACTLY what a few shops in our area did and they diversified in different directions. The gaping holes were spices, homebrew, foods and other things for non-Brits and organic basics such as flour and vegetables and meat.
The reasons that the big boys avoid these things are because (1) they have to buy for the whole of the UK and they only want to deal with giant vendors that can sell X-hundreds of tons of the stuff. They can't buy a box of this and a packet of that. (2) Certain things only appeal to a few people and these are not spread out evenly across the UK. (3) Some products (homebrew) are politically unacceptable as they may be seen to encourage unwelcome behaviour. (4) Some products are not easily available at all times (e.g. locally slaughtered organic meat) and a giant chain must be seen to have the same things on offer at all times.
So one shop right opposite a huge Morrisons now has
homebrewing kit and supplies as their number one selling range, as well as
organic baking supplies including baking tins for breads and cakes and all the bits and pieces that go with that stuff. That is about 150 sq m of floor space just for that.
Another became a
spice and health-food store and sells spices in kilo and 5kg bags and organic unbleached flours in 1kg and 15kg bags. They evolved into a small retail front shop of about 30 sq m and a 200 sq m wholesale unit for restaurants and smaller shops throughout the district.
One became a
deli and sells anything and everything that non-British Europeans want like
real bread, sauerkraut, condiments that the Brits don't know (e.g. Maggi, Jamaican sauces) and
freshly roasted coffee beans (which also lead to a healthy sale of coffee grinders and other coffee apparatus. The smell of coffee beans roasting drew people in from across the street!
So that is what small shops that were lucky enough to have a giant open up right next to them did.
They used the opportunity to their advantage.
BUT be warned, you (a) have to get it right and get your lines of supply in place before you do anything - and (b) it's an all-or-nothing deal. The old usual wholesale rubbish has to go and
in all cases, the owners renamed and rebranded their shops. Wholefood, health-food, deli, farm-shop - those seem to be the keywords.