This is the process that's beginning to happen with Stories Highlights, a tool Instagram launched this past December that lets you save past Stories to your profile. Many users categorize these Highlights: There might be one collection of food-related Stories posts, another for shopping, and a third for travel. For each Highlight, you have an option to select a cover image, which will appear on your profile, in a small circle above your feed posts.
The default for this cover image is the Story in that Highlight that you posted first. So, if you're including Story posts from January 5 and 6, the first one to go up on January 5 would appear as your cover photo, although you can always select another Story post from the Highlight to take its place.