Disclaimer: I am not a lawyer, this is not legal advice. This is a recounting of my own experience with the US immigration system and the solutions I've found from friends and acquaintances. If you are a faculty member thinking about how you can support students, please skip to the last section.

I wish more faculty were up front about this during my PhD, so I'll be up front: as far as I know, there are really four (...let's say 3.5) ways of staying in the US as a non-American PhD student in the humanities and social sciences:

1. Get sponsored by your employer, whether it be a university or an organization outside of academia. This is "what's supposed to happen," but lol at the academic job market, and realistically the only orgs outside of academia who can do this in a reasonably expeditious way are big tech companies or consulting firms. Of course, you then risk your department turning on you for "selling out" while they also decline to support you in any way in your job search.
2. Get married to a US citizen and have enough money / resources to apply for a green card. In some ways, this is the most straightforward approach, but also one that presumes a great deal of privilege (wealth being a huge one).
3. Get your department to apply for STEM OPT designation if you are in the social sciences. This doesn't entirely solve your problem since your OPT still expires, but this buys you much more significant time (2 years) to find an employer sponsor.
4. Apply for an EB1 (extraordinary ability) green card yourself. This also presumes a great deal of privilege (similar to 2, given the amount of money you'll have to pay up front), and it involves a lot of documentation that is really above and beyond.

My partner and I tried options 1 and 2. The first option was basically impossible, partially because going on the job market is such a painful and precarious experience, but more importantly, even making sure that he could stay in the country to do an MIT postdoc was a fight that we only resolved by getting married. For option 2, we did it as sparsely as we could -- no lawyers, got a professor friend to translate materials, DIY'd lots of legal docs, didn't get the thing that would allow Stephan to travel -- and even then it still cost thousands of dollars and many headaches. Being on medical leave and then reading for quals, all while Stephan was on the job market and the green card stuff was up on the air...that was a pretty frenetic time in my life.

The one thing we did learn throughout this process is that USCIS does not actually run on taxpayer dollars, so when the government shut down in December 2018, our green card process was not actually affected, and we finished the initial round of green card processing in about 5 months, which felt like a breeze compared to what we were preparing for. Pursuing option 3 via the reclassification policy from ICE requires a great deal of advocacy and collective action and many unions (like UE, which MIT is affiliated with) do not have specialized immigration lawyers. In my experience, international student offices mostly exist to ensure that universities are in compliance with immigration regulations with the least cost possible to the institution, often to the detriment of student welfare.

Option 4, applying for EB1, is an option that we had not considered (and didn't know about at the time), but given the examples I'll share, may require some creative thinking. Razvan Marinescu has written extensively about his process applying for EB1 himself, where he has also generously shared the entire application (and LaTeX template!) he used so that folks can learn from the language and structure. Marinescu is an extraordinarily talented scientist, no doubt -- but I'd be curious to hear if others have had experience doing this in fields where the scholarly conventions are a bit different. That all being said, I do think this is an excellent point of departure for folks to liberally draw from grant and fellowship applications that have required you to explain time and again what your research is about and why it matters.

What can I do as a faculty member?

1. Help students get department designation changed to STEM if possible. This is by far the biggest thing that would be nothing short of transformative for every international student in your department. Having two extra years of OPT doesn't negate all of the bullshit that international students go through just to apply for OPT, but this would be a game changer.
2. If it is appropriate, offer to write recommendation letters and to facilitate department production of documents that help the immigration process along. Many of these letters are not tall orders -- they simply attest to the fact that this student is in fact a student and is a talented researcher.
3. If you are a US citizen, do not justify or explain the cruelty of the US immigration system to international students. They know. It is their reality. They confront it every day. You should be the one listening to them, since the likelihood that you missed something or don't understand the implications for international students is rather high.
4. If you were an international student yourself, do not tell students that if you did it, so could they. This is up there with "the job market rewards people who try," "you'll be fine," and "everything will work out." Do not invalidate your students' experiences, particularly since immigration and the academic job market has changed a great deal even year to year. Ask how you can help, and proceed accordingly.
5. Do not denigrate Big Tech or corporate jobs. Yes, McKinsey and Meta aren't exactly saving the world and have questionable ethical principles, but they're generally also going to pay PhD graduates well and give them labor protections and benefits that universities never have. Generally speaking, they are also the most well equipped to employ PhD graduates as the application pipelines for these jobs are much more streamlined, structured, and predictable (e.g., recruiting seasons with scheduled resume drops). Referrals and networking certainly helps -- and this by no means detracts from how difficult it is still to find a job -- but many of these processes are reasonably well-documented (there are a lot of case prep books!) compared to networking your way into a small niche. This cannot be said for more specialized jobs that "make direct use of your research," as if that is simply the bare minimum for a job they might deign to be interested in.