This is a self-answer. I spent some time thinking, did some googling, and then some more thinking.
If her only magnetic ability was to attract, then no, she couldn't
An electromagnet (or any kind of magnet- for that matter) can only attract and cannot repel a magnetic metal.
I'm not sure how I didn't realize this when I first thought of the question, since I played with magnets quite a bit when I was a kid. Somehow that experiential intuition failed me. In my defense, I don't have a lot of science/physics background, and I thought that electromagnets could possibly be a special case in interactions with magnetic metals. They aren't.
Attraction between objects and herself isn't her only magnetic ability
At the time that I first wondered-up what is now the above question post, I was very fixated on the logic of the magnetic attraction ability shown in the clip that is the subject of the question. I had somehow "forgotten" about this.
As stated in the question post (referencing the steel plating in Railgun S ep8 at 7:29), Misaka can also move objects around magnetically without there being any "equal and opposite force" exerted upon herself. There's also the stairs at 10:32, the entire floor at 13:17, and an army of dolls in ep10 at 5:30.
Additionally, Misaka seems to be able to choose which steel object she attracts herself to (or to herself) in her vicinity. That facility she was in had a lot of steel in it (she hints toward that herself during the same battle with Frenda at 10:23: "If you wanted to make me fall, you'd have to rebuild this entire facility and this time, leave out the steel!"), presumably enough that the specific steel cage she used to escape to wasn't the only steel object nearby. In that same clip, there were some pipes to her left, and she had just shielded herself opposite to the steel she escaped to with some steel plating, which looked to be much closer to herself than the steel cage. If she could only create "mutual" attraction between objects and herself, she probably would have only been able to draw herself to the ground (the closest thing to herself) in that scenario.
When she moves the Gekota finger puppet with the hairclip inside it to stand up in her hand and twirl around, that probably wouldn't be possible if her only magnetic ability were to attract objects to herself if the hairclip doesn't have a permanent magnetic field. But if it actually had a permanent magnetic field, maybe that would be possible?
You can read more about her electromagnetic abilities on the Toaru wiki.
There's also the possibility of usage of magnetic (eddy current) braking.
To my memory, she is never shown using such a "magnetic repel" ability (which- as already mentioned- would be scientifically bunk) in the anime's three seasons. I don't know about the manga or the light novels.
She is shown using her ability to create magnetic attraction between herself and other magnetic metals plenty of times in the anime: season 1 ep8 at 16:58, season 2 ep1 at 17:08, ep9 at 5:15, and ep12 at 12:20- just to name a few, and particularly with this cage escape clip and ep9 at 5:15, it's apparant that she can choose what she attracts herself to- even if other magnetic metal objects are closer by.
So she probably could have softened the impact to the cage by attracting herself to something opposite to her direction of movement before impact.
But that would kind of defeat the story, making it less satisfying.
In that scene, the damage done to herself serves a storytelling purpose: she's in a pinch against Frenda, and eventually the rest of the members of ITEM by herself, and having very hard time handling it. She was trying to solve everything independently. That's in stark contrast to the situation in ep23-34 after her character growth in ep21.
It's science fiction
Her ability to create magnetic attraction between objects and herself certainly seems more believable to me than the other magnetic abilities she has (a force acting between herself and another object rather than a force acting between an object and seemingly nothing), but I guess that at that point it's a bit pointless to question it too deeply, and one should just accept that it's science fiction created for entertainment value, and to just enjoy the entertainment value. It's the mixture of believable-enough abilities and limitations (like the self-damage of hitting a steel cage) that makes the story interesting and enjoyable. It's probably better to just let the story work its magic (no pun intended) to bring about immersion and some suspension of disbelief.