This work addresses the sensitivity of the algorithm proposed by Andrieu and Doucet (IEEE T. Signal Proces., 47(10), 1999), for the joint Bayesian model selection and estimation of sinusoids in white Gaussian noise, to the values of a certain hyperparameter claimed to be weakly influential in the original paper. On the basis of extensive numerical experiments, we argue on the contrary that the value of this hyperparameter (the scale parameter of the expected signal-to-noise ratio) has a strong influence on 1) the mixing rate of the Markov chain and 2) the posterior distribution of the number of components. Fully Bayesian and Empirical Bayes methods are proposed and compared for estimating an appropriate value for this hyperparameter. In the Fully Bayesian approach, a weakly informative proper prior is assigned over that hyperparameter. In the Empirical Bayes approach, marginal likelihood maximization is performed by means of an importance sampling-based Monte Carlo EM (MCEM) algorithm. The pros and cons of each method are discussed on the basis of numerical experiments conducted with several sample sizes and signal-to-noise ratios.