If new phenomena beyond the Standard Model will be discovered at the LHC, the properties of the new particles could be determined with data from the High-Luminosity LHC and from a future linear collider like the ILC. We discuss the possible interplay between measurements at the two accelerators in a concrete example, namely a full SUSY model which features a small (tau) over tilde -LSP mass difference. Various channels have been studied using the Snowmass 2013 combined LHC detector implementation in the Delphes simulation package, as well as simulations of the ILD detector concept from the Technical Design Report. We investigate both the LHC and the ILC capabilities for discovery, separation and identification of various parts of the spectrum. While some parts would be discovered at the LHC, there is substantial room for further discoveries at the ILC. We finally highlight examples where the precise knowledge about the lower part of the mass spectrum which could be acquired at the ILC would enable a more in-depth analysis of the LHC data with respect to the heavier states.