Vern—had the exact same thing happen to me too—only difference was the half digested fish was an eel-pout.
Dan—I’m told the trick to circle hooks is to not set the hook—you just gently lift the rod tip and the hook is supposed to roll and turn and bite the corner of the mouth.
I have been fishing for more than 60 years—I can not seem to just ‘not set the hook’.
I drift fish big brookies here with dead smelt in the river—I missed so many strikes I quit using circle hooks but I did hook a couple as you describe in the side.