It is well-known that the surface tension of small droplets and bubbles deviates significantly from that at the planar interface. In this work, we analyze the leading corrections in the curvature expansion of the surface tension, i.e., the Tolman length and the rigidity constants, using a "hybrid" square gradient theory, where the local Helmholtz energy density is described by an accurate equation of state. We particularize this analysis for the case of the truncated and shifted Lennard-Jones fluid, and are then able to reproduce the surface tensions and Tolman length from recent molecular dynamics simulations within their accuracy. The obtained constants in the curvature expansion depend little on temperature, except in the vicinity of the critical point. When the bubble/droplet radius becomes comparable to the interfacial width at coexistence, the critical bubble/droplet prefers to change its density, rather than to decrease its size, and the curvature expansion is no longer sufficient to describe the change in surface tension. We find that the radius of the bubble/droplet in this region is proportional to the correlation length between fluctuations in the liquid-phase.