Slice with overlapping

What a lovely idea. 🙂

I would recommend opening up the image, duplicating that image for as many sections that you want to turn into separate images, masking them, and then exporting them as separate images. Here's how:

1. Open the image.
2. Select the Background layer and go to Layer > New > Layer Via Copy (you can also duplicate the layer with the keyboard shortcut Ctrl (Windows) / Command (macOS) + J).
3. Repeat Step #2 for as many images that you want to make from the image.
4. Select the Rectangular Marquee tool (if you're not sure where to find it, you can use Edit > Search or click the Magnifying glass icon in the upper right to search for it. Click on it in the results and Photoshop will show you where it is and select it for you).
5. In the Layers panel, select the first duplicated layer above the original Background layer.
6. Make a select around the portion of the image that you want to keep (Draw a rectangular selection by dragging from the upper left to the lower right.).
7. While the selection is active, make a layer mask by clicking on the 3rd icon from the left at the bottom of the Layers panel.
8. Repeat Steps 5-7 for each duplicated layer.
9. Go to Layer > Export As... 
10. In the Export As dialog, deselect any layers on the left side that you don't need to export (probably just Background), on the right under File Settings, set the format (try PNG so that you don't have JPEG artifacts), and click Export.