  We study the critical gravitational collapse of a massless scalar field
non-minimally coupled to gravity, using a quadratic coupling function with a
strength parameter $\xi$. We concentrate on critical phenomena of type II, and
determine with an accuracy of at least $10^{-12}$ the value of the critical
amplitude for collapse to a black hole, as well as the values of the critical
and echoing exponents. Obtaining such high accuracy in the critical amplitude
requires us to do a coordinate radial transformation that effectively increases
resolution near the central regions by a factor of at least $10^3$. As
expected, we find that for the case of small coupling the critical behaviour is
very similar to that of a minimally coupled scalar field. On the other hand,
for high coupling the dynamics become so violent that we need to introduce a
special slicing condition, known as the shock-avoiding slicing condition, in
order to avoid gauge pathologies that would otherwise cause our simulations to
fail. With this new gauge condition we are able to perform high accuracy
simulations even in the strong coupling regime, where we find that the critical
and echoing exponents become significantly smaller, and that the echoing
behavior is richer and can not be modelled by a single harmonic.
