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 m c o e x . It is argued that in the limit L->~ an unconventional first-order transition occurs at a characteristic value m t <m c o e x , where a large equilibrium droplet ceases to exist, and the thermodynamically conjugate variable to m, the magnetic field H, exhibits a jump from H t ( 1 ) to H t ( 2 ) . It is found that H t ( 1 , 2 ) scale like L - d / ( d + 1 ) their ratio being simply H t ( 1 ) /H t ( 2 ) =(d+1)/(d-1), and m c o e x -m t L - d / ( d + 1 ) as well, while the excess thermodynamic potential (relative to its value according to the double-tangent construction) varies as g t 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 H t is rounded over an interval Δm L - d 2 / ( 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.