Just as the question asks: Are there other physical entrances to/exits from Diagon Alley aside from the little yard in the back of the Leaky Cauldron? ETA to clarify the question: physical entrances.
Sure. Floo powder, at the very least. I'm sure one could apparate into it (though possibly just the bar?). The vanishing cabinet, after Spring 1997 (end of Harry, et. al.'s sixth year). A dragon, though that was more an exit than an entrance. As for walk-in entrances, I'm pretty certain there are no others mentioned in the books. Every time the Leaky Cauldron entrance is mentioned, it's referred to as "the" entrance. And it's probably hard enough to hide from muggles as-is, how may muggle-proof entrances can they realistically create and maintain?
I'm fairly certain that Hagrid, when he freed Harry from Knockturn Alley, used some other way to get him back to Diagon (though it's not 100% clear from the book exactly how):
Hagrid seized Harry by the scruff of the neck and pulled him away from the witch, knocking the tray right out of her hands. Her shrieks followed them all the way along the twisting alleyway out into bright sunlight. Harry saw a familiar, snow-white marble building in the distance — Gringotts Bank. Hagrid had steered him right into Diagon Alley. (src: Chamber of Secrets).
“Yer a mess!” said Hagrid gruffly, brushing soot off Harry so forcefully he nearly knocked him into a barrel of dragon dung outside an apothecary.
So, the entrance is near an apothecary, and some disance from Gringotts, though a twisting alleyway.
I'm not aware of a mention of any other entrances that are directly physically accessible by land (Kevin already covered teleport-type and flying routes).