Recent research has described an improved method of Fowler–Nordheim (FN) plot analysis, based on the definition and evaluation of a slope correction factor and a new form of intercept correction factor. In this improved approach, there exists a basic approximation that neglects certain terms in the general theory, and focuses on the influence of the form of the tunneling barrier on the values of basic slope (σB) and intercept (ρB) correction factors. Simple formulae exist that allow these to be evaluated numerically for a barrier of arbitrary well-behaved form. This paper makes an initial exploration of the effects of barrier form on FN plot analysis. For a planar emitter, two models for the correlation-and-exchange (C&E) potential energy (PE) are used. For the Schottky–Nordheim barrier, it is shown that numerical and analytical approaches generate equivalent results. This agreement supports the validity of the numerical methods used. Comparisons with results for the Cutler–Gibbons barrier show that small differences in the assumed C&E PE make little difference to values of σB and ρB. Schottky's planar image PE has then been used, in conjunction with the electrostatic PE variation associated with a spherical emitter model, to explore the influence of apex radius ra on correction-factor values, for values of ra ≥ 20 nm. Both σB and ρB increase significantly as ra decreases, especially ρB. At low values of barrier field F, σB depends approximately linearly on 1/F, with a slope that depends on ra. Suggestions are made for how the exploratory work described in this paper might be extended.