Nope, woodcock is NOT the only exception.
I stand corrected, as per the act below. I will rephrase, in Ontario, the only migratory bird you can hunt with lead shot is woodcock.
http://laws-lois.justice.gc.ca/eng/regulations/C.R.C.,_c._1035/page-7.html#docCont
15.1 (1) Subject to subsection (2), no person shall
(a) possess, for the purpose of hunting a migratory game bird, except a woodcock, band-tailed pigeon or mourning dove, shot other than non-toxic shot; or
(b) hunt a migratory game bird, except a woodcock, band-tailed pigeon or mourning dove, by the use of shot other than non-toxic shot.
(2) The exceptions of woodcock, band-tailed pigeon and mourning dove set out in subsection (1) do not apply in
(a) any area in which the use of non-toxic shot is required under Schedule I when hunting those species; and
(b) the wildlife areas set out in Schedule I to the Wildlife Area Regulations.
SOR/90-623, s. 1; SOR/93-432, s. 2; SOR/97-400, s. 2; SOR/2013-126, s. 2.