TO "bail out" has two meanings: to remove water from a vessel (usually by hand and often as a desperat emeasure); and to free another from prison by paying their "bail money". Hay is not involved.

I suspect, subject top further research, that the two terms come from the same root structure, probably akin to bailiff, also.

Etymonline says:

bail (v.)
1613, from baile "bucket" (1336), from O.Fr. baille "bucket," from M.L. *bajula (aquae), lit. "porter of water," from L. bajulare "to bear a burden" (see bail (n.1)).

bail (n.1)
"bond money," 1485, developed from "temporary release from jail" (1466), and that from earlier meaning "captivity, custody" (1259). From O.Fr. baillier "control, guard, deliver," from L. bajulare "to bear a burden," from bajulus "porter," of unknown origin.

So it seems I was right--the underlying concept being "bear a burden", bajulare.

A bailiff also comes from the Latin bajulus, porter or bearer of burdens, as does his jurisdiction, a bailiwick. A "bailey", related as well, is a wall enclosing an inner court, but its origin is not quite clear.