Add an index on a new column
No, sorry.

PostgreSQL generally indexes NULL fields, so would it would have to add an entry for each row.  You can make an index not index NULLs (using a partial index, or using a non-default type of index which doesn't index NULLs), but it is not implemented for such an index to observe an all-NULL constraint on the column and so take a shortcut of not scanning the table.  It would uselessly scan the table looking for not-NULLs anyway.  Interesting idea though.