Given the limited space I'm not sure that doing it it chunks will meet "quick deadlines" and I would pursue adding a new filegroup and a new file to it using ALTER DATABASE ADD FILE. Then, create the unpivoted table using SELECT INTO and specify the new filegroup using the ON clause.

-- Create a new filegroup and add a file to it
ALTER DATABASE [database_name]
ADD FILEGROUP [filegroup_name];

ALTER DATABASE [database_name]
ADD FILE (
  NAME = [file_name],
  FILENAME = 'C:\path\to\filegroup\file.ndf',
  SIZE = 100GB,
  MAXSIZE = UNLIMITED,
  FILEGROWTH = 10GB
)
TO FILEGROUP [filegroup_name];

-- Create the temporary table on the new filegroup
SELECT *
INTO [filegroup_name].[unpivoted_table]
FROM (
  SELECT id, col1, col2, col3, col4, col5, col6, col7, col8, col9, col10, col11
  FROM origin_table
) AS t
UNPIVOT (
  value FOR col IN (col1, col2, col3, col4, col5, col6, col7, col8, col9, col10, col11)
) AS unpvt
ON [filegroup_name];

-- Optional: Rename the origin table
EXEC sp_rename 'origin_table', 'origin_table_backup';

-- Copy the unpivoted table to where the origin table is normally stored
SELECT *
INTO origin_table
FROM unpivoted_table;

-- Drop the backup table, once the oigin_table is verified
-- DROP TABLE origin_table_backup;

Obviously this can be broken into separate steps to suit, and handling both a backup table and the new table in the currently limited space may not be possible so you may want to delay overwriting "origin_table" until you have resolved the current lack of space. Until you have the origin_table rebuilt perhaps use the unpivoted_table in lieu.