In this paper, novel convergence accelerating techniques are adapted onto the proposed hyperbolic numerical inverse Laplace transform method (hyperbolic-NILT) and analyzed. This ID NILT method is based on the approximation of the inverse kernel of the Laplace transform Bromwich integral exp(st). It is shown that with the use of the convergence accelerating algorithms onto the essence of the proposed NILT method, an enhancement on the core of the inversion is achieved, with relatively accurate and stable results, while preserving valuable time and memory. The algorithms are tested and their corresponding results are discussed, mainly regarding the accuracy, stability and computational efficiency. The experimental accuracy analysis tests are implemented in the universal MATLAB language with properly chosen Laplace transforms.