Step 1: The question describes a lateral support running from the cervix and upper vagina out to the side wall of the pelvis - this is the classic description of the cardinal ligament.
Step 2: The cardinal (transverse cervical) ligament of Mackenrodt is a thickening of pelvic fascia sitting at the cervical level, enclosing the uterine artery and providing the main lateral anchorage of the uterus.
Step 3: Check the other options. The broad ligament is only a peritoneal sheet, the round ligament heads to the labium majus, and the pubocervical ligament runs forward to the pubic bone - none go laterally to the pelvic wall.
Step 4: The structure that fits the lateral cervix-to-pelvic-wall path is the transverse cervical ligament.
\[\boxed{\text{Transverse cervical ligament}}\]