High-resolution birefringence (Δn) measurements are carried out to probe the critical behavior at the smectic-A-smectic-C (Sm-A-Sm-C) phase transition in a binary system. The critical behavior of this transition is explored with the aid of a differential quotient extracted from the Δn values. The results obtained reveal that the Sm-A-Sm-C and nematic-smectic-A (N-Sm-A) transitions exhibit nonuniversal behaviors with effective exponents lying between the tricritical and three-dimensional XY values and follow two distinctly different curves with decreasing width of the Sm-A and N phases, respectively. The origin of such critical behavior is a unique feature for the respective phase transitions.