I haven't looked at the caselaw in Texas, but based on my experience that is consistent with what I have seen trial lawyers say about it online is that:
(1) generally speaking, the phone was "discoverable" - that means it has potentially relevant information on it and can be requested with a proper discovery request in this particular litigation;
(2) the plaintiffs here did indeed request the phone or downloads from the phone of messages and media but Jones refused this request (I'm not sure the history on whether the plaintiffs moved the court to compel its production);
(3) the rule is where material is discoverable and property requested, it is then upon the party that holds the material to seek protection from the court in the form of an exclusion from production where the material is privileged, and also it is typical for parties with potentially sensitive information (that is not actually privileged) to get a protective order that requires the other side receiving the material can only use it for the case and cannot publish it or use it for any other purpose. We routinely use protective orders in this way . . . but Jones's lawyers did not seek a protective order in the case.
(4) when Jones's lawyer accidentally produced the entire digital contents of the phone to the plaintiffs, the plaintiffs' attorney did what he's supposed to: he notified counsel of the mistake and gave counsel an opportunity to attempt to "claw back" information that should not have been produced, but Jones's attorney did not respond. Texas has a "claw back" period and after a certain number of days, plaintiffs counsel was permitted to presume that Jones had no further objections to the material being in plaintiffs' custody.
(5) at that point, the cat is out of the bag. The plaintiffs' lawyer is in lawful possession of the material and if he doesn't want to give it back, he doesn't have to. And because there's no protective order, he is not constrained on providing it to others - though I think he's being smart and only providing it upon receipt of a subpoena because that further covers his arse from claims by Jones.
(6) there still may be a basis for Jones to keep some of the material confidential - but these grounds are very narrow. He's already gotten the judge to rule that information about Jones's medical history and treatment are confidential. And we can presume that truly privileged material will be protected as well, but the only solid privilege there is would be attorney-client - and so that would only protect material between Alex and his lawyers. I suppose he could try to assert some kind of journalistic privilege but I suspect that's not going to be persuasive.
I think the horse if out of the barn and while the plaintiffs' lawyer is being smart about it, that phone is going to be produced to the J6 Committee and potentially other investigations including DOJ. Whether the contents actually make it to the public sphere remain to be seen but that's highly possible.