John Ryan's try had put the Munstermen ahead in the first half, while the boot of Ian Keatley kept their Scottish hosts at bay.
Hidalgo-Clyne earned the chance to snatch a dramatic victory after Will Helu finished off a flowing move for his first Edinburgh try, but it wasn't to be.
"It was a superb try and I thought we could get in there at the end," said Solomons. "But I think we just had to hold the ball down there and the points would have come.
"There's no question about that - we should have won this game.
"I thought we played poorly against Zebre [two weeks ago] and fully deserved to lose. I thought we should have come away with a draw at Connacht [last weekend], and this game is a game we should have won.
"We've got to look at ourselves. It's those little things, those fine margins we spoke about at the end of the game between winning and losing, and we've got to make sure those fine margins fall on our side."
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