  The quantum triangular XXZ model has recently enjoyed a wealth of new
theoretical results, especially in relation to the modeling of the
Ba$_3$CoSb$_2$O$_9$ compound. In particular, it has been understood that in a
longitudinal magnetic field the umbrella (cone) phase, classically stable in
all the easy-plane region of the ground-state phase diagram, is considerably
reduced by the effect of quantum fluctuations. We provide more quantitative
information for this phenomenon at arbitrary value of the site spin $S$, by
employing the dilute Bose gas expansion, valid in the high-field regime; our
results improve the available estimates of the $1/S$ expansion. We quantify the
extent to which a higher spin suppresses the effect of quantum fluctuations.
Besides, we show how in three-dimensional layered systems a relatively small
antiferromagnetic interlayer coupling has a similar consequence of bringing
back the umbrella phase in some part of the phase diagram.
