What you want is a camera with the smallest f-number and the largest sensor. DSLRs do well, but that would be hard to do on your budget. I could possibly recommend some used DSLR stuff but it would be right at the edge of your budget and not be available everywhere.
These will all be a few years old so that they are in your budget.
For a single all-in-one, look at the Panasonic Lumix DMC-LX100. It has a very large sensor (this one is called m4/3); it splits the difference between a DSLR with an APS-C sensor and the typical high end compact with a 1" sensor. New it is just over 600, used for 450. At the wide end of the zoom range it is f/1.7
Close but not as good is the Canon PowerShot G1 X Mark II. It as a microscopically larger sensor. (1mm larger in each direction). It is f/2.0 on the wide end. New it is $650, used 350-400.
Also consider the sony rx100 series. They have several models. They are a slightly smaller, but still impressively large sensor (1"). Prices vary from 350 to over 1000 depending on model. The lenses are all different from year to year, but tend to have small f-numbers. Because of the smaller sensor this model won't do as good as say the LX100, but this one will fit in a small pocket (like jeans) while the LX100 is larger and more obvious.
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