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: