It's possible to move components between the template and a page.

1. Click the component you want to move and hold the left mouse button.
2. Drag it to the top of the page to the Edit Page or Edit Template tab.
3. Wait until you see that you have switched mode and place the component where you want.

Tip: You can also copy components over different pages. Select the component you want to copy and type Ctrl/Command + C. Go to the page where you want to move the component to and type Ctrl/Command + V.

Related articles:

How do I add a component to a page in Website Builder?

Which templates can I use in Website Builder?