The smectic-A/air interface of liquid-crystal droplets with antagonistic boundary conditions is studied by atomic force microscopy (AFM). The droplets are prepared on coated silicon wafers on which a planar alignment is preferred in contrast to the homeotropic alignment at the air interface. As a result, focal conic defects appear in the smectic-A phase causing a characteristic pattern of depressions in the droplet surface. The dimensions of the defect-induced depressions are measured by AFM as a function of temperature for two different compounds possessing a smectic-A-isotropic and a smectic-A-nematic transition. Whereas the results are independent of temperature in the smectic-A-isotropic case, reflecting the first-order nature of the transition, a pronounced temperature dependence is observed for the second compound, where the depth of the defect-induced depressions decreases continuously with increasing temperature and vanishes at the second-order transition to the nematic phase. These observations can be qualitatively explained through the behavior of the layer compressional elastic constant at the smectic-A-nematic transition.