Can one dictate the order of generated values when adding a new BIGSERIAL column to a table?

I need to add a new BIGSERIAL column to a huge table (~3 billion records). This question is similar to what I need to do and the accepted answer has helped me somewhat. But I'm still wondering about something. In my case, the table already has a BIGSERIAL column which is the primary key, but many rows have been deleted so now there are *gaps*. (The table has subsequently been fully vacuumed.) I need to regenerate the values so that they are sequential again. Here are 5 example rows of what I want to achieve where the new_value > 1000:

+---------+---------+
|old_value|new_value|
+---------+---------+
|1026     |1001     |
|1027     |1002     |
|1030     |1003     |
|1032     |1004     |
|1039     |1005     |
+---------+---------+

I have successfully implemented the alternative approach as mentioned in the referenced answer above (CREATE TABLE ... and then INSERT INTO new_table SELECT * FROM ... ), but I would also like to attempt, and benchmark against, the initial suggestion. The problem, however, is that I don't know whether the new_value will be generated **in the same order** as the old_value as this is a requirement. How can I ensure the order of the new_value column follows/tracks the order of the old_value column when the new_value column is added using a statement like this:

ALTER TABLE existing_table ADD COLUMN new_value BIGSERIAL;

## A different approach I also attempted the following (that works quite well on a small table), but it's much slower than the alternative suggestion of the referenced answer on very large tables:

ALTER TABLE existing_table ADD COLUMN new_value BIGINT NOT NULL DEFAULT 0;

UPDATE existing_table AS existing
SET new_value = generated.new_id
FROM (
      SELECT original.old_value
           , row_number() OVER (ORDER BY original.old_value) AS new_id
      FROM existing_table AS original
     ) generated
WHERE existing.old_value = generated.old_value;