We consider the problem of efficient, anytime-valid statistical inference of the Average Treatment Effect (ATE) in a sequential experiment where the assignment of subjects to treatment or control can be made adaptively over time. We relax assumptions necessary forasymptotic normality of the Adaptive Augmented Inverse-Probability Weighted (A2IPW) estimator introduced by Kato et al. (2021), which is semiparametrically efficient due to the adaptive assignment. With the aim of enabling continuous data analysis as the data is being collected, we then derive both asymptotic and nonasymptotic confidence sequences that are considerably tighter than previous methods. In addition to sharper inference tools, we use propensity score truncation techniques from the recent off-policy estimation literature to reduce finite sample variance of our estimator without affecting asymptotic variance (which is optimal). Empirical results demonstrate that our methods yield narrower confidence sequences that maintain time-uniform error control.