What's the difference between at hand, on hand and in hand? At hand seems to me as if you have something in reach. On hand is if you have something in stock. And in hand can be used as if you have more time at your disposal.
The way these different prepositions are used is all very much idiomatic, so there will be a limit to how far you can "rationalise" why any particular preposition is associated with any particular meaning. Especially in respect of "hand", since most of us always have a couple of those about our person ready to be incorporated into our word choices.
One thing I would say that "in" usually denotes something closer than "on" (which itself is usually closer than "at"), which is why the "nearby" meanings use the latter two.
Note that "in hand" doesn't actually mean having more time at your disposal - it's just that we often say we have "time in hand" (things in hand can be in reserve, spare). Apart from the literal held in the hand, the other common idiomatic usage for this one is that if something is "in hand" it's being actively dealt with (by implication, really close to you), not just sitting on your "to-do" list.
OP doesn't mention to hand, which is actually more common than the others in British English (but seemingly quite rare for Americans). In practice few would choose at hand or on hand on the basis of how physically close/accessible something was - largely pointless anyway, since few others would apprehend any subtle distinction being made. Go British and make life easier for everyone!
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At hand frequently refers to proximity in time, the other two usually relate to spatial proximity:
With the apocalypse at hand, I was pleased to find I have a cache of food and water on hand, and a rifle in hand.
Think of measures of distance:
in hand - literally within one's grasp - not even inches away - "A bird in the hand"
at hand - nearby - a few feet away at most - "He kept his cigarettes at hand"
on hand - somewhere accessible but fairly distant - more than a few feet away and less than a few hundred yards - "We have more vacuum cleaners on hand"