Wanted: Small bike for the girlfriend

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If it's a lady specific frame then you can go higher. My mum at 5ft 1 on a good day (& 60) rode a Raleigh Pioneer hybrid & Peugeot racing bike, both were 20"/51cm frames.
If she's reasonably able to swing her leg & doesn't want a proper step through granny bike then I'd go by floor to saddle height as an indicator as to frame sizes available, you can then adjust bar height/reach easily enough.
That way a lot more frames become available to you.
If at all possible get her to sit on a bike so that she feels comfortable height wise. Measure from top of saddle to floor (I'd guess at about 34-35") and distance from nose of saddle to centre of bars.
All you need do then is ask prospective sellers if the saddle will go down to that height and compare reach to the bars, (you can buy shorter or longer stems to acommadate).
It's a foolproof method.
HTH