Finding a decent camera in that size is damned near impossible, but something like a LUMIX ZS60S or a Sony DSCW830 is pretty tiny and takes much nicer photos than a phone once you've spent a little time getting used to the camera and what it can actually do. (Often your phone can also take much better photographs than your automatic settings will, but fiddling with it is a pain.)
There are plenty of superzooms (which are basically a mirrorless or DSLR without an interchangeable lens) that cost too much and get you too little and are too big, but the compact market is weird and tiny and not great.
Sony's current compact line seems OK, though I haven't used them--the DSCWX cameras. They've got Exmor sensors (their mirrorlesses all use Exmor sensors) and reasonable lenses. You won't get mirrorless performance, but you're not expecting to, so, yeah.
