We use 5-brane webs to study the two-dimensional space of supersymmetric mass deformations of higher rank generalizations of the 5d E1 and E∼1 theories. Some of the resulting IR phases are described by IR free supersymmetric gauge theories, while others correspond to interacting fixed points. The number of different phases appears to grow with the rank. The space of deformations is qualitatively different for the even and odd rank cases, but that of the even (odd) rank E1 theory is similar to that of the odd (even) rank E∼1 theory. One result of our analysis predicts that the supersymmetric SU(N) theory with CS level k = N/2 + 4 and a single massless antisymmetric hypermultiplet exhibits an enhanced global symmetry at the UV fixed point, given by SU(2) × SU(2) if N is even, and SU(2) × U(1) if N is odd.