We apply a new entropic scheme to study the critical behavior of the square-lattice Ising model with nearest-and next-nearest-neighbor antiferromagnetic interactions. Estimates of the present scheme are compared with those of the Metropolis algorithm. We consider interactions in the range where superantiferromagnetic (SAF) order appears at low temperatures. A recent prediction of a first-order transition along a certain range (0.5-1.2) of the interaction ratio (R = Jnnn/Jnn) is examined by generating accurate data for large lattices at a particular value of the ratio (R = 1). Our study does not support a first-order transition and a convincing finite-size scaling analysis of the model is presented, yielding accurate estimates for all critical exponents for R = 1 . The magnetic exponents are found to obey "weak universality" in accordance with a previous conjecture.