Cycling jackets don't have hoods because you have a bike helmet, on your head.
Get a helmet rain cover.
and with a hood attached, turning your head inside the hood won't help you see behind you,
a separate one will turn with your head...
a cycling rain cape gives coverage, over your arms and thus, feet,
and can use waterproof fabrics,
because they have airflow ventilation on the underside..
but the speed you ride will be slower, not racing..
....