We consider the Schramm-Loewner evolution (SLE_κ) for κ∈(4,8), which is the regime that the curve is self-intersecting but not space-filling. We let K be the set of κ∈(4,8) for which the adjacency graph of connected components of the complement of an SLE_κ is a.s. connected, meaning that for every pair of complementary components U,V there exist complementary components U₁,…,U_n with U₁=U, U_n=V, and ∂U_i∩∂U_i+1≠∅ for each 1≤i≤n−1. It was proved by Gwynne and Pfeffer [11] that this set is non-empty. We show that the range of an SLE_κ for κ∈K is a.s. conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an SLE_κ for κ∈(4,8) and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter.