Perhaps from the door post on which tavern accounts were maintained, hence an account or score.

From The Bard's Comedy of Errors (1590) we have:

I pray you, jest, sir, as you sit at dinner. I from my mistress come to you in post; If I return, I shall be post indeed For she will scour your fault upon my pate. Methinks your maw, like mine, should be your clock, And strike you home without a messenger.

Perhaps from "the door post on which tavern accounts were maintained, hence an account or score" (quoted from OED Compact print edition).

From The Bard's Comedy of Errors (1590) we have:

I pray you, jest, sir, as you sit at dinner. I from my mistress come to you in post; If I return, I shall be post indeed For she will scour your fault upon my pate. Methinks your maw, like mine, should be your clock, And strike you home without a messenger.