Sampling from the posterior probability distribution of the latent states of a hidden Markov model is nontrivial even in the context of Markov chain Monte Carlo. To address this, Andrieu et al. (2010) proposed a way of using a particle filter to construct a Markov kernel that leaves the posterior distribution invariant. Recent theoretical results have established the uniform ergodicity of this Markov kernel and shown that the mixing rate does not deteriorate provided the number of particles grows at least linearly with the number of latent states. However, this gives rise to a cost per application of the kernel that is quadratic in the number of latent states, which can be prohibitive for long observation sequences. Using blocking strategies, we devise samplers that have a stable mixing rate for a cost per iteration that is linear in the number of latent states and which are easily parallelizable.