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 $\widehatλ_c$ which is smaller or equal to the classical critical percolation intensity $λ_c$ and derive sharp conditions for $\widehatλ_c>0$ 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 $\widehatλ_c<λ_c$.