Searches for continuous gravitational waves from unknown neutron stars are limited in sensitivity due to their high computational cost. For this reason, developing new methods or improving existing ones can increase the probability of making a detection. In this paper we present a new framework that uses Markov chain Monte Carlo (MCMC) or nested sampling methods to follow up candidates of continuous gravitational-wave searches. This framework aims to go beyond the capabilities of (which is limited to the sampler), by allowing a flexible choice of sampling algorithm (using as a wrapper) and multidimensional correlated prior distributions. We show that MCMC and nested sampling methods can recover the maximum posterior point for much bigger parameter-space regions than previously thought (including for sources in binary systems), and we present tests that examine the capabilities of the new framework: a comparison between the , , and samplers, the usage of correlated priors, and its improved computational cost. Published by the American Physical Society 2024