A new shock-tracking technique that avoids re-meshing the computational grid around the moving shock-front was recently proposed by the authors. This paper describes further algorithmic improvements which make the extrapolated Discontinuity Tracking Technique (eDIT ) capable of dealing with complex shock-topologies featuring shock-shock and shock-wall interactions. Various testcases are included to describe the key features of the methodology and prove its order-of-convergence properties.