How to implement Ola Hallengren backup plan on the server for some databases nightly full backups and for other weekly full backup only
No you don't need to install the Ola's solution again since the initially installed commandexecute.sql and databasebackup.sql would have been saved on master DB as SP.

You just need to create a new job , as one created early but this time for 5 left over DB;s and schedule it weekly per you're timings.

EXECUTE dbo.DatabaseBackup
@Databases = 'USER_DATABASES',-- Name of those 5 DB's would come here
@Directory = 'C:\Backup',-- You're backup path
@BackupType = 'FULL',
@Verify = 'Y',
@Compress = 'Y',
@CheckSum = 'Y'



You can use above script while creating the T-SQL job and schedule it accordingly.

More details on how to do above refer to schedule job

Also, just to make sure both, initial backup and this newly created backup job not to run on same time you can put a condition or per the analysis when you're daily backup job completes, you can schedule this new weekly/nightly backup job for remaining 5 DB's