Following the 2013-14 outbreak in French Polynesia, the Zika virus (ZIKV) epidemic spread widely to many countries where Aedes Aegypti as the main transmitting vector is endemic. The lack of a second wave of ZIKV infection in most affected regions may suggest that a sufficiently high level of herd immunity was reached during the first wave. We developed an agent-based transmission model to investigate the role of asymptomatic infection on the likelihood of observing a second wave, while accounting for its relative transmissibility. We found that, as the relative transmissibility of asymptomatic infection increases, a second wave is more likely to occur, despite an increase in the attack rate during the first wave. When the reproduction number varies between 1.9 and 2.8 based on estimates for Antioquia, Colombia, the attack rate varies between 4% and 26% for a low (below 10%) effectiveness of interventions in blunting the ZIKV transmission from symptomatic cases to mosquitoes. Moreover, the fraction of cases due to sexual transmission is estimated below 4% of the cumulative incidence. Our analyses underscore the need to quantify the transmissibility of asymptomatic infections, without which the overall attack rates and the level of herd immunity cannot be accurately estimated.