Well, I must agree that I haven't encountered it used to mean 'at another time' in the sense of a distant time.

Here's another example from William Morris's "The Well at the World's End":

"... because there was no man that beheld her,
but anon he was the thrall of her love, and might not pluck
his heart away from her to do any of the deeds whereby men
thrive and win the praise of the people."

Anon refers to an indefinite time, but not too far ahead, the exact length of time depending on the context. I dare say one would expect that Shakespeare's servant would come to hand more quickly than Morris's swain would fall in love. (Mind you, I have felt myself growing older while waiting to be served in some restaurants.)