Looks like I found the perfect way to get those project pages to show the correct breadcrumbs.
Before, each project page was listed under the Projects section, but when you clicked on those pages, there was no breadcrumb trail leading back to that section. Some Googling I did didn't seem to shed any light onto it.
And then I stumbled across the Custom Breadcrumbs module.
The pages aren't very clear, but in short, it lets you define custom breadcrumbs for particular content types, e.g. my project content type.
After I installed the Custom Breadcrumbs module, it was a simple matter of defining the new breadcrumb trail for the project content type, and that was it.
Hooray!