When you split a package, rename each resulting new package as described in Renaming a Single Package. If one of the resulting new packages is not renamed, the pre-split and post-split versions of that package are not compatible and might violate dependency logic when the end user tries to update the package.
Rename the original package, and include require dependencies on all new packages that resulted from the split. This ensures that any package that had a dependency on the original package will get all the new pieces.
Some components of the split package can be absorbed into existing packages as a merge. See How to Enable Your Application to Use a Shared Area.