A phenomenological theory of phase coexistence of finite systems near the coexistence curve that occurs in the thermodynamic limit is formulated for the generic case of d-dimensional ferromagnetic Ising lattices of linear dimension L with magnetization m slightly less than mcoex. It is argued that in the limit L→∞ an unconventional first-order transition occurs at a characteristic value mt< mcoex, where a large equilibrium droplet ceases to exist, and the thermodynamically conjugate variable to m, the magnetic field H, exhibits a jump from Ht(1) to Ht(2). It is found that Ht(1,2) scale like L- d/( d+1) their ratio being simply Ht(1)/ Ht(2)=( d+1)/( d-1), and mcoex- mt∝ L- d/( d+1) as well, while the excess thermodynamic potential (relative to its value according to the double-tangent construction) varies as gt∝ L-2 d/( d+1) . The prefactors in all these relations are derived and it is shown that near the bulk critical point this transition shows a standard scaling behavior and the prefactors can be expressed in terms of known universal constants. Also the rounding of this transition at very large but finite L is considered and it is found that the jump in H at Ht is rounded over an interval ∆ m∝ L- d2/( d+1) . Various simulations are interpreted in the light of these predictions, and the possibility to extract the surface free energy of liquid droplets coexisting in a finite volume with supersaturated gas is critically discussed.