In this paper, we present a hybrid algorithm for the reconstruction of compressively sensed signals sparse in the Hermite transform (HT) domain. The aim is to combine the advantages of two reconstruction approaches: the gradient algorithm well-known for its wide range applicability, and the recently proposed highly efficient Hermite coefficient tresholding algorithm. The later is based on the theoretically derived signal support detection threshold, which takes into account the specific properties of the observed sparsity domain. Reducing the compressive sensing noise level and increasing the component coefficients values with a partial time-domain reconstruction, the gradient algorithm prepares the signal for the tresholding procedure.