This paper presents a first step of our research on designing an effective and efficient GP-based method for symbolic regression. First, we propose three extensions of the standard Single Node GP, namely (1) a selection strategy for choosing nodes to be mutated based on depth and performance of the nodes, (2) operators for placing a compact version of the best-performing graph to the beginning and to the end of the population, respectively, and (3) a local search strategy with multiple mutations applied in each iteration. All the proposed modifications have been experimentally evaluated on five symbolic regression benchmarks and compared with standard GP and SNGP. The achieved results are promising showing the potential of the proposed modifications to improve the performance of the SNGP algorithm. We then propose two variants of hybrid SNGP utilizing a linear regression technique, LASSO, to improve its performance. The proposed algorithms have been compared to the state-of-the-art symbolic regression methods that also make use of the linear regression techniques on four real-world benchmarks. The results show the hybrid SNGP algorithms are at least competitive with or better than the compared methods.