To this I will add that the Hall effect must also be taken into account
somehow; I don't know how! This off-axis loop problem is shmutzig; could
the original poster please tell us why the problem is of interest.
This question came up when my AP Physics class was examining a problem
from the 1996 E&M AP C level exam. That problem specifically asked the
students to indicate the direction of the induced E field. One of my very
bright students asked me about the non-coaxial loop of wire and I just
couldn't answer him. It doesn't seem to me to be a transient situation.
It seems that, as long as I keep the loop in that position there will be a
current in the loop that seems to oppose the direction of the induced E
field predicted by the coaxial loop and Lenz's Law. Thanks to all of
those who have responded so far.