Can I set the exact sending time that a follow-up message is sent?

The short answer here is no. 

We've worked really hard to keep Mailshake very easy to use while still providing a lot of power. While we may revisit this feature suggestion, at the moment we feel it would add unnecessary friction, and for certain scenarios, it would make the sending calendar and sending rules quite confusing. 

The best way for you to control when your messages go out is to adjust your schedule settings with a time window and to block off certain days.