An efficient method for Bayesian inference in stochastic volatility models uses a linear state space representation to define a Gibbs sampler in which the volatilities are jointly updated. This method involves the choice of an offset parameter and we illustrate how its choice can have an important effect on the posterior inference.A Metropolis-Hastings algorithm is developed to robustify this approach to choice of the offset parameter. The method is illustrated on simulated data with known parameters, the daily log returns of the Eurostoxx index and a Bayesian vector autoregressive model with stochastic volatility.