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1.
On the third hand -- used to describe an alternative or option which comes to mind (or to hand) *after* the one you just thought of and described as OTOH (on the other hand). Frequently but not always, the OTTH option would steer you back in the direction you were going with your first thought.

Sholom Aleichem's creation Tevye the Milkman (or Dairyman), better known these days as the main character in "Fiddler on the Roof", not infrequently came up with arguments back and forth when he was trying to decide something -- he could have used a fourth, fifth, or sixth hand sometimes.
"Are you going to keep on going out with him?"

"Well, I like him well enough, and we have a lot in common. OTOH, he is a bit high-maintenance. But hey, OTTH, I have some time and money available right now to invest in looking for a good long-term relationship . . . so yeah, I think I will, and we'll see where it goes from here."

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A paraphrase of a famous description by Dr. Teeth (leader of the Electric Mayhem) of his drummer Animal: "When he gets ahold of something, he could hit it. OTOH, he could eat it. OTTH -- well, there is no OTTH."
by jalp March 20, 2011