The Oxford English Dictionary deals, under the meaning of hand, with prepositional phrases associated with it. And there are literally dozens of them. It categorises them by the particular preposition included with the word hand, category a) being those associated with at, b) being those with by c) being for one's own etc.
Category i) deals with phrases involving on with hand, and there are many of those. They are further sub-divided into a number of sub-categories. I believe that category i), sub-categories d,e,f,and g deal with the matter you have raised in your question - or at least have some bearing upon it.
In order to illustrate the extent of the Pandora's box opened up by the question I will list those sub-category meanings with some examples:
(d) on all hands (also on every hand): on all sides, in all
directions, to or from all quarters
Examples are available from 1540, but the most recent one is:
1990 G. Gilder Life after Television (1992) 22 33 Contrary to the
rich and variegated promise of new technology proliferating options on
every hand, TV squeezes the consciousness of an entire nation.
this first one d) is not directly related to your question, but see e) below:
(e) on (the) one hand (also †on one hand): used to introduce a point
of view, fact, case, etc., followed by another which (typically)
contrasts with it, introduced by on the other (hand). on the other
hand: used to introduce a contrasting point of view, fact, case, etc.,
typically following another introduced by on (the) one hand.
e) is the straightforward on the one hand..., on the other hand... to which you refer. There are examples from 1581, this being the most recent:
2011 Independent 17 Mar. (Viewspaper section) 2/3 The OBR's
forecast implies relatively strong job creation. The OECD's, on the
other hand, implies continued misery for hundreds of thousands.
f) deals with on either hand - still recognising that most people have only two hands.
†(f) on either hand: on either side, in either case (now rare). Also
†on some hands (also †on this hand): in some cases, in this case
1999 P. R. Brenner in J. D. Davidson & K. J. Doka Living with Grief
vii. 87 This new model must transcend the either/or alternatives
that drive practice to extremes on either hand.
Finally g) deals with on any hand.
(g) on (also †upon) any hand: on any account, in any case; cf. at any
hand at Phrases 1a(g).
Examples date from 1600, the most recent:
2010 A. Vladimirov et al. Assessing Information Security i. 41 On
any hand, it is clearly required to verify both completeness and
correctness of any follow-up reaction to its predecessor.
So in answer to your question, whilst I cannot find any OED examples which specifically refer to another hand, the fact is that this expression is both historically and currently variegated. And many examples exist at d) and g) which seem to go beyond the recognition of there being only two hands available. It would seem therefore that you would not be departing too far from accepted precedent by employing the term another hand.
Indeed Oxford English Dictionary may be glad to hear from you about the another hand examples which you have turned up.