According to Phrase Finder the expression "mad as a March hare" dates back to the 16th century and refers to the breeding season of hares.
Is there evidence of a different or earlier origin of this saying?
As Jim notes in a comment above, Etymology Online reports that
Phrase mad as a March hare is attested from 1520s, via notion of breeding season;
Christine Ammer, It's Raining Cats and Dogs... and Other Beastly Expressions (1989) goes Etymonline one better by finding a related expression in Chaucer:
Older than all of these [hare-related] terms is the saying mad as a March hare, which dates from at least 1500 and possibly much earlier (about 1385 Chaucer wrote, "This somnour wood [mad] were as an hare..."). The most famous colorfully crazy March Hare was the one that joined in the mad tea party in Lewis Carroll's Alice in Wonderland, where it buttered the Mad Hatter's watch to make it run again, then dipped it into his tea, and remarked, "It was the best butter, you know."
Ammer also discusses at some length the habits of hares in breeding season:
It was long believed that hares are unusually wild in March, during the rutting season. According to Erasmus, their behavior could be accounted for by the absence of hedges and other cover at that time of year; but Erasmus mistook the saying for "mad as a marsh hare," somehow confusing the month with a swamp. At any rate, during rutting season, it was reported, hares boxed each other's ears with their clublike forepaws and kicked at each other with their strong hind legs, exhibiting, it was concluded, normal mating behavior in which the males fight each other over females. In the 1970s some British naturalists undertook a study of hares in Somerset, ... Mating among hares takes place at night and not just in March, but from January through August. ... Moreover, the embattled hares are not males fighting against males, but females fighting against males, presumably to resist their advances. Since the females are generally bigger than males, they often win these battles. Nor do the fighting partners end their battles in mating; the triumphant female simply hops away.
Ammer's quotation from Chaucer is from near the beginning of "The Friar's Tale" in The Canterbury Tales. The friar begins his tale by painting a picture of an avaricious summoner working hand in hand with an equally venal arch-deacon:
He [the "erchedeken"] hadde a Somnour redy to his hond,/A slyer boy was noon in Egelond;/For subtilly he hadde his espiaille/That taughte him, wher that him might availle./He could spare of lechours onn or two,/To techen him to foure and twenty mo./For thogh this Somnour wood were as an hare,/To telle his harlotrye I wol nat spare;/For we been out of his correccioun;/They han of us no Iurisdiccioun,/Ne never shullen, terme of alle hir lyves.
In the context given, wood does indeed mean "mad."