Once a SQL Server database is online then there is no option to append additional data changes from a backup to that 'online' database.
Having said that there is one way you could try to apply more changes to that database. To do this you will have to take this database offline by changing its state to 'norecovery' (using restore database with 'norecovery' option). After this you can apply further backups to this same database. Now the backups that 'could' be applied will need to have the correct LSN sequence (so if the next backup was was a FULL then that will need to be applied, if it was a DIFF then that will need to be applied etc).
I hope this helps in your situation.