Sampling from a probability density constrained to a manifold is of importance in numerous applications arising in statistical physics, statistics or machine learning. Sampling from such constrained densities, in particular using an MCMC approach, poses significant challenges and it is only recently that correct solutions have been proposed. The resulting algorithms can however be computationally expensive. We propose a relaxation of the problem where the support constraint is replaced with that of sampling from a small neighbourhood of the manifold. We develop a family of bespoke and efficient algorithms adapted to this problem and demonstrate empirically their computational superiority, which comes at the expense of a modest bias.