Looks to me like the standard "barrel" or "panel" style jack which is commonly available from guitar parts suppliers. You just fixed the physical floppiness so I think you're good to go. Going by the other components on my Gibson EB-5, they probably used a quality jack, not a flimsy budget one, so probably no need - or way - to upgrade. (There was a more expensive make with redundant plug contacts. I tried to order one once and it was out of stock, now they're not even available.)
This style jack isn't my favorite but it should be reliable enough for a while - a lot of higher end makers use it.
If the electronic connection becomes intermittent in the future, you will need to replace it. Switchcraft is a trusted brand - I would avoid anything cheaper.
I have the 5-string version of your bass from 2013 with the top mounted "standard" jack. There is a certain "cool" factor but it's not all gravy; the pickguard and/or screws can get strained over time if you're not careful unplugging. After using both styles a bunch, over the years I've made my peace with side mounted barrel jacks. In your situation, I would not bother trying to source a 2013 pickguard (and plugging the old side hole) just for the sake of the jack.
I suppose if you really wanted to convert to the standard style jack (which allows you to manually bend the contacts) you could ream out the side more and install a "football" or cup-style jack plate. (The Electrosocket is pretty beefy.) I do prefer that style but I don't think it's worth the trouble to convert.
HTH
