Attended Home Delivery, where customer attendance at home is required, is an essential last-mile delivery challenge, eg, for valuable, perishable, or oversized items. Logistics service providers are often faced with the absence of their customers. In this paper, we consider the delivery problem in which customers can be revisited on the same day by a courier in the case of a failed first delivery attempt. Specifically, customer presence uncertainty is considered in a two-stage stochastic program, where penalties are introduced as recourse actions for failed deliveries. We build on the notion of a Customer Availability Profile (CAP) defined as a profile containing historical time-varying probability information of successful deliveries. We tackle this stochastic program by developing an efficient parallelized ALNS solution algorithm. Our results show that by achieving a right balance between increasing the hit rate and reducing travel cost, logistics service providers can realize costs savings as high as 32% if they plan for second visits on the same day.