Probabilistic parameter estimation in model fitting runs the gamut from maximum likelihood or maximum a posteriori point estimates from optimization to Markov Chain Monte Carlo (MCMC) sampling. The latter, while more computationally intensive, generally provides a better characterization of the underlying parameter distribution than that of point estimates. However, in order to efficiently explore distributions, MCMC methods ideally require generating uncorrelated samples while also preserving reasonable acceptance probabilities; this becomes particularly important in problematic regions of parameter space. In this paper, we extend a recently proposed Hamiltonian MCMC sampler parametrized by neural networks (L2HMC) by modifying the loss function to jointly optimize the distance between samples and the acceptance probability such that it is stable and efficient. We apply this enhanced sampler to parameter estimation in a recently proposed MRI model, the multi-echo spherical mean technique. We show that it generally outperforms the state of the art Hamiltonian No-U-Turn (NUTS) sampler, L2HMC, and a least squares fitting in terms of accuracy and precision, also enabling the generation of more informative parameter posterior distributions. This illustrates the potential of machine learning enhanced samplers for improving probabilistic parameter estimation for medical imaging applications.