Another skinny foot/narrow heel person here! I have just one pair of flats that stay on - they have an elastic across the top of my foot, sort of like a cross between a ballet flat and a Mary-Jane style shoe. The only other flats I can wear are loafers, with a higher vamp.
I don't really wear pumps at all. Heeled booties in fall/winter/early spring. I can wear heeled sandals in summer if they have lots of straps across the top to hold my feet in place.
My go-to brands are Ecco in a 38 and Clark's sized down a half-size (7.5). My actual shoe size is an 8, but in these two brands, slightly smaller than expected. Rieker also works, but in a 39 and only in certain styles (closed shoes).
I once went into a big dept. store and tried on all kinds of different brands of shoes, and made a list of what brands worked, and in what size. Unfortunately, I lost the list, so am going to have to do this again. There were some brands that didn't work at all on my feet, no matter what the size was. In the brands that did work, there was no real consistency as to whether a 7.5 or an 8 was my 'true' size. When measured, my foot is a size 8. Half of the brands I tried that fit were 7.5 and the other half were 8's.
I recommend trying on a whole bunch of different shoes over the coming weeks/months. Keep a list going. Maybe you only try two or three 'new' brands at a time, but make sure you jot down what works. That way, when you spy a great sale somewhere, you know if the brand works and what size to order.