Can you access the auto increment value in MySQL within one statement?

try this

INSERT INTO users (default_assignment) VALUES (LAST_INSERT_ID()+1);

there's no need to create another table, and max() will have problems acording to the auto_increment value of the table, do this:

CREATE TRIGGER trigger_name BEFORE INSERT ON tbl FOR EACH ROW
BEGIN
   DECLARE next_id;
   SET next_id = (SELECT AUTO_INCREMENT FROM information_schema.TABLES WHERE TABLE_SCHEMA=DATABASE() AND TABLE_NAME='tbl');
   SET NEW.field = next_id;
END

I declare the next_id variable because usually it will be used in some other way(*), but you could do straight new.field=(select ...)

CREATE TRIGGER trigger_name BEFORE INSERT ON tbl FOR EACH ROW
BEGIN
   SET NEW.field=(SELECT AUTO_INCREMENT FROM information_schema.TABLES WHERE TABLE_SCHEMA=DATABASE() AND TABLE_NAME='tbl');
END

Also in cases of (SELECT string field) you can use CAST value;

CREATE TRIGGER trigger_name BEFORE INSERT ON tbl FOR EACH ROW
BEGIN
   SET NEW.field=CAST((SELECT aStringField FROM information_schema.TABLES WHERE TABLE_SCHEMA=DATABASE() AND TABLE_NAME='tbl') AS UNSIGNED);
END

(*) To auto-name an image:

SET NEW.field = CONCAT('image_', next_id, '.gif');

(*) To create a hash:

SET NEW.field = CONCAT( MD5( next_id ) , MD5( FLOOR( RAND( ) *10000000 ) ) );