The requirement to separate topographical effects from surface electrochemistry information is a major limitation of scanning electrochemical microscopy (SECM). With many applications of SECM involving the study of (semi)conducting electrode surfaces, the hybridisation of SECM with scanning tunnelling microscopy (STM) or a surface conductance probe would provide the ultimate topographical imaging capability to SECM, but previous attempts are limited. Here, the conversion of a general scanning electrochemical probe microscopy (SEPM) platform to facilitate contact electrical conductance (C)‐ and electron tunnelling (T)‐SECM measurements is considered. Measurements in air under ambient conditions with a Pt/Ir wire tip are used to assess the performance of the piezoelectric positioning system. A hopping‐mode imaging protocol is implemented, whereby the tip approaches the surface at each pixel until a desired current magnitude is exceeded, and the corresponding z position (surface height) is recorded at a set of predefined xy coordinates in the plane of the surface. At slow tip approach rates, the current shows an exponential dependence on tip‐substrate distance, as expected for electron tunnelling. For measurements in electrochemical environments, in order to overcome well‐known problems with leakage currents at coated‐wire tips used for electrochemical STM, Pt‐sensitised carbon nanoelectrodes are used as tips. The hydrogen evolution reaction on 2D Au nanocrystals serves as an exemplar system for the successful simultaneous mapping of topography and electrochemical activity.