Vaccination has been proposed as one of the most effective methods to combat the COVID-19 pandemic. Since the day the first vaccine, with an efficiency of more than 90%, was announced, the entire vaccination process and its possible consequences in large populations have generated a series of discussions on social media. Whereas the opinions triggered by the administration of the initial COVID-19 vaccine doses have been discussed in depth in the scientific literature, the approval of the so-called 3rd booster dose has only been analyzed in country-specific studies, primarily using questionnaires. In this context, the present paper conducts a stance analysis using a transformer-based deep learning model on a dataset containing 3,841,594 tweets in English collected between 12 July 2021 and 11 August 2021 (the month in which the 3rd dose arrived) and compares the opinions (in favor, neutral and against) with the ones extracted at the beginning of the vaccination process. In terms of COVID-19 vaccination hesitance, an analysis based on hashtags, n-grams and latent Dirichlet allocation is performed that highlights the main reasons behind the reluctance to vaccinate. The proposed approach can be useful in the context of the campaigns related to COVID-19 vaccination as it provides insights related to the public opinion and can be useful in creating communication messages to support the vaccination campaign.