In SharePoint 2013, if you create a new subsite and select Use the top link bar from the parent site, there is no way to go to the home page without opening another browser. This blog post will give you a tip on how to add a home link on the Content Editor Web Part with HTML.
1. Create a new subsite. Under Use the top link bar from the parent site? select No.
2. In the subsite, edit the page and add a Content Editor Web Part (found in Media and Content) under Categories.
3. Next, modify this Web Part and click Edit Source on the Ribbon as shown below.
4. Add the following code to the content.
5. Click OK and launch.
You can change the URL for the link to suit your needs. You can also edit the text on the page, but this is a quick solution for your home site navigation.