The countability is a red-herring.
The OED classes a free hand as a phrase:
P5. to have a free hand (also to have one's hands free): to have the freedom to act completely at one's own discretion. Also with a free hand, with one's hands free; similarly to give a free hand.
1838 Times 5 Jan. 6/2 If..Lord Mulgrave had been sent out to Canada with his hands free, and with unfettered power to carry into effect his own wise and benevolent views.
1989 AJ 28 June 15/3 The council..had wisely given the community group a fairly free hand in converting the building.
2002 Independent 9 Apr. 8/2 The photographer..was recruited by Luciano Benetton..in 1983 with a free hand to sell the brand.
As such, it is set and not really capable of being analysed any more than "He gave me carte blanche."
To add to this, it is probably an idiom - there is not much sense in the literal version - what can you do when one hand is not "free"? It is hardly the complete freedom that the phrase/idiom implies.