Sql-server – Ola Hallengren scripts: after a job failure, how to skip the databases that worked and backup only the remaining ones

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

Most likely it is a 3rd party backup solution such as NetBackup that your sysadmins have pointed to your instance.

Try reading the default trace to see if the application name column gives you the specifics:

-- read all available traces.
-- including current
DECLARE @current VARCHAR(500);
DECLARE @start VARCHAR(500);
DECLARE @indx INT;

SELECT @current = path
FROM sys.traces
WHERE is_default = 1;

--remove the rollover number from the filename 
--(i.e., log_157.trc becomes log.trc)
SET @current = REVERSE(@current)
SELECT @indx = PATINDEX('%\%', @current)
SET @current = REVERSE(@current)
SET @start = LEFT(@current, LEN(@current) - @indx) + '\log.trc';

-- CHANGE FILTER AS NEEDED
SELECT 
    TextData,
    ApplicationName, 
    StartTime,
    CASE EventClass
        WHEN 46 THEN 'Object:Created'
        WHEN 47 THEN 'Object:Deleted'
        WHEN 164 THEN 'Object:Altered'
        ELSE 'Other'
    END as EventClass, 
    DatabaseName, 
    ObjectName,
    ObjectType, -- https://msdn.microsoft.com/en-us/library/ms180953.aspx
    HostName, 
    LoginName
FROM
    ::fn_trace_gettable(@start, DEFAULT)
WHERE 1=1
    AND DatabaseID <> 2 --eliminate tempdb
    AND DatabaseName = 'model'--Like '%master%'
    AND Textdata like 'Backup Database%'
ORDER BY StartTime DESC