A low-autocorrelation binary sequences problem with a high figure of merit factor represents a formidable computational challenge. An efficient parallel computing algorithm is required to reach the new best-known solutions for this problem. Therefore, we developed the $\mathit{sokol}_{\mathit{skew}}$ solver for the skew-symmetric search space. The developed solver takes the advantage of parallel computing on graphics processing units. The solver organized the search process as a sequence of parallel and contiguous self-avoiding walks and achieved a speedup factor of 387 compared with $\mathit{lssOrel}$, its predecessor. The $\mathit{sokol}_{\mathit{skew}}$ solver belongs to stochastic solvers and can not guarantee the optimality of solutions. To mitigate this problem, we established the predictive model of stopping conditions according to the small instances for which the optimal skew-symmetric solutions are known. With its help and 99% probability, the $\mathit{sokol}_{\mathit{skew}}$ solver found all the known and seven new best-known skew-symmetric sequences for odd instances from $L=121$ to $L=223$. For larger instances, the solver can not reach 99% probability within our limitations, but it still found several new best-known binary sequences. We also analyzed the trend of the best merit factor values, and it shows that as sequence size increases, the value of the merit factor also increases, and this trend is flatter for larger instances.