Even amid the unprecedented public health challenges attributable to the COVID-19 pandemic, opposition to vaccinating against the novel coronavirus has been both prevalent and politically contentious in American public life. In this paper, we theorize that attitudes toward COVID-19 vaccination might "spill over" to shape attitudes toward “post-pandemic” vaccination programs and policy mandates for years to come. Drawing from a large, original panel study of American adults conducted during the pandemic, a novel survey experiment and two observational surveys, we find this to be the case. Specifically, we observe evidence of spillover onto general vaccine skepticism, flu shot intention, and attitudes toward hypothetical vaccines (i.e., vaccines in development), which do not have pre-existing attitudinal connotations. Attitudes toward childhood vaccines - a well-known and established set of vaccines - see less consistent spillover effects. Further, these spillover effects vary by partisanship and COVID-19 vaccination status, with the political left and those who received two or more COVID-19 vaccine doses becoming more pro-vaccine, while the political right and the unvaccinated have become more anti-vaccine. Taken together, these results point to the salience and politicization of the COVID-19 vaccine impacting non-COVID vaccine attitudes. We end by discussing the implications of this study for effective health messaging.