Rather than having to rerun an entire analysis, it would be really useful to only enrich certain features.
My original dilemma, for which there is no solution:
After enriching my layer, it informed me that five of the polygons weren't enriched due to geometry areas. Since it wasn't too many, I went through and just manually drew them in and added the data from the existing shapefile. However, I now need to enrich those five features, and since it's enrich layer rather than enrich feature, I'm not sure how to manage that. Sure, it's not very hard to just delete the extra columns, but with over 800 polygons it was an expensive analysis the first time around, and I do NOT want to get charged for that again when I only need five more. I tried fiddling around with the current extent, but I either get extra features, or it doesn't pick up the ones I'm zoomed in on.
Since Kelly Gerrow (unfortunately) confirmed my suspicion that there was no effective way to handle that, I'm taking her advice to post this as an idea.