The integration of shared e-scooters with public transit is a promising solution for urban mobility's first/last-mile challenge. This study explores spatiotemporal factors influencing this integration, using 35-day e-scooter trip data from Chicago. Employing a random-effect negative binomial approach, we modeled the frequency of e-scooter trips to access/egress to/from bus stops and train stations. Results indicate that weather conditions, design features like intersection density, and multimodal network density significantly influence usage. The transit system characteristics such as service frequency have a positive effect on the integration of e-scooters and trains while a similar effect for bus and e-scooter integration was not significant. Furthermore, safety-related variables such as accident and crime rates as well as demographic characteristics were also revealed to be significant factors in our study. These findings offer vital insights to urban planners and policymakers for infrastructure, safety enhancements, and interventions to encourage efficient e-scooter-public transit integration.