In this paper we consider a well-known construction due to Gromov and Lawson, Schoen and Yau, Gajer, and Walsh which allows for the extension of a metric of positive scalar curvature over the trace of a surgery in codimension at least 3 to a metric of positive scalar curvature which is a product near the boundary. We extend this construction for (p,n)-intermediate scalar curvature for 0≤p≤n−2 for surgeries in codimension at least p + 3. We then use it to generalize a well known theorem of Carr. Letting Rsp,n>0(M) denote the space of positive (p,n)-intermediate scalar curvature metrics on an n-manifold M, we show for 0≤p≤2n−3 and n ≥ 2, that for a closed, spin, (4n − 1)-manifold M admitting a metric of positive (p, 4n − 1)-intermediate scalar curvature, Rsp,4n−1>0(M) has infinitely many path components.