  We derive a sufficient condition for the existence of a subcritical
percolation phase for a wide range of continuum percolation models where each
vertex is embedded into Euclidean space according to an iid-marked stationary
Poisson point process. In contrast to many established models, the probability
of existence of an edge may not only depend on the distance and the weights of
its end vertices but also on a surrounding vertex set. Our results can be
applied in particular to models combining heavy-tailed degree distributions and
long-range effects, which are typically well connected. More precisely, we
study the critical annulus-crossing intensity $\widehat{\lambda}_c$ which is
smaller or equal to the classical critical percolation intensity $\lambda_c$
and derive sharp conditions for $\widehat{\lambda}_c>0$ by controlling the
occurrence of long edges. We further present tail bounds for the Euclidean
diameter and number of points of the typical cluster in the subcritical phase
and apply our results to several examples including some in which
$\widehat{\lambda}_c<\lambda_c$.
