A finger clock, I use a mitutoyo one they are less fragile than the rest I have tried.
A monkey, basically an arm that will fit in a collete/chuck in the spindle and allow the finger clock to move out and clock round parts. it's hard to explain/I am rubbish at explaining
A 0.25mm / 0.01" feeler I use this for setting tools directly on a job
An edge finder like your bleepy one is fine (I made an inverted h one with a magnet as an aprentice that allows you to clock an edge.
A roller with a dowel in it, think of a ground plate with a magnet on it (rubbish if your not working in ferrous metals) with a dowel attached this allows you to get a position on angled surface.
My discriptions are a bit rubbish but it's what I have used for the past 17 years.