I'm not convinced that even with finite free rides public transit users would make rush hour worse. The people taking the tube at rush hour are usually doing it where they cannot travel at an alternate, less congested time. Those people will go for free or not, and are stuck riding at the times they do. Even with a free ride card, you would still want to avoid rush hour just to not have to wait for transit to become available.
What is interesting is that most public transit obeys the same congestion rules as roads - if you take a congested road system and add more roads, you often just end up with more cars on the roads at the same times congesting the new roads again because of all the offset load that uses transit at off hours due to congestion. For people on the roads, they are effectively "free" to use once you have the car, but people still actively decide to drive at off hours to avoid congestion. The same happens with any transit system regardless of cost - unsurprisingly, people don't want to be stuck in traffic.
What is interesting is that most public transit obeys the same congestion rules as roads - if you take a congested road system and add more roads, you often just end up with more cars on the roads at the same times congesting the new roads again because of all the offset load that uses transit at off hours due to congestion. For people on the roads, they are effectively "free" to use once you have the car, but people still actively decide to drive at off hours to avoid congestion. The same happens with any transit system regardless of cost - unsurprisingly, people don't want to be stuck in traffic.