Basta adicionar o '0' no seu select.
INSERT INTO table_name (a,b,c,d)
SELECT
other_table.a AS a,
other_table.b AS b,
other_table.c AS c,
'0' AS d
FROM other_table
INSERT INTO table_name (a,b,c,d)
SELECT
other_table.a AS a,
other_table.b AS b,
other_table.c AS c,
'0' AS d
FROM other_table