Is it possible to send a SIGTERM signal before connection draining when deploying a new task definition in ECS?

We have an application that requires clients to have long lasting socket connections, and have code in place application side to gracefully handle a SIGTERM event. It does depend on connections remaining established but my observation is that the SIGTERM happens after the de-registration delay has elapsed and thus all active connections are killed.

Is there a mechanism by which the SIGTERM can be sent before connection draining starts, or perhaps some other signal that would otherwise tell us that the application instance will soon be terminated?