How to snip same part of screen as last time?
Greenshot has an option to Capture last region. Highly recommended.
Also very easy to do quick edits such as highlighting and obfuscation:
As far as I know there is no way to "re-snip the previously snipped area" using the snipping tool.
The best way to achieve a similar functionality with the snipping tool is to the Window Snip or Full-Screen Snip available in the Mode Menu. Clicking the same window showing a different section of code etc. and trimming it down to the section you want.
Alternatively the open source tool ShareX looks to have a similar functionality as you describe. The menu options for that are further described here in a GitHub feature request for said functionality.
ProTip 1: The keys Shift+Win+S launches the new/alternative (?) snipping tool in Windows 10 quicker.
ProTip 2: Windows 10 in 1903 now has a multi-clipboard build in, turn it on in the settings, and you can use Win+V to paste any of the last 10 (?) copied items.
One fairly straightforward way to do this is to separate the capture operation from the crop operation. Capture either the entire screen with Ctrl+PrnScr or a window with Alt+PrnScr; paste into an image editor which supports layers1; repeat until you have all of the captures as layers of a single image; then crop the image and save the layers out one by one.
1 I use the GIMP, but I'm sure there are plenty of other options.