Best way to add a new column with an initial (but not default) value?

To add the column with a default and then delete the default, you can name the default:

ALTER TABLE tbl ADD col INTEGER NOT NULL CONSTRAINT tbl_temp_default DEFAULT 1
ALTER TABLE tbl drop constraint tbl_temp_default

This filled in the value 1, but leaves the table without a default. Using SQL Server 2008, I ran this and your code, of alter update alter and did not see any noticeable difference on a table of 100,000 small rows. SSMS would not show me the query plans for the alter table statements, so I was not able to compare the resources used between the two methods.

I'd ALTER TABLE tbl ADD col INTEGER CONSTRAINT tempname DEFAULT 1 first,, and drop the explicitly named constraint after (presumably within a transaction).

You can do it in an insert trigger

Another, maybe more native, way would be:

ALTER TABLE tbl ADD COLUMN col INTEGER NOT NULL DEFAULT 1;
ALTER TABLE tbl ALTER COLUMN col DROP DEFAULT;

I'm not sure how long this function exists, but the PostgreSQL documentation goes back to version 7.1 and for 7.1 it is already described.