Specific-national visas, like for Canadians, Australians, Chileans and Singaporeans,
H-1B visas, which have laxer requirements but are subject to a 1-in-3-ish lottery for for-profit orgs. It's uncapped for research-related non-profits!
If you're not one of the listed nationals and you don't check the O-1 boxes, the H-1B is the next best option. The lottery happens only once a year, at the end of March, and if you win it's another six months before you can start work.
This means that if you want to work for a US organisation and you're happy to try your luck, it's a good idea to start interviewing at those organisations around about now.
Be aware that small, nimble organisations can sign you up for the lottery within a week of making an employment offer, but larger organisations or ones with less experience with immigration might take much longer.
(This is written on the basis of only my own experience of the US immigration system; you'll want to do a lot more research while evaluating your own options. The exactness of the H-1B timing came as a surprise to me, and maybe this'll help other folk)
Thanks for this post Andy two important clarifications
Although the lottery is in 6 weeks, to get the paperwork prepared to apply can still require some time. Ideally, you should start interviewing at least a few months before to give companies the time to evaluate candidates, make a decision, and then process the paperwork and consult with the lawyers. So applying now is pushing it pretty close to the deadline. EA organisations may be a bit quicker than the average US company.
Not all nonprofits are H1-B cap exempt. Basically: non-profits which are not research organisations or affiliated to them are not cap exempt. According to this website, only the following organisations are cap exempt:
Institutions of higher education;
Non-profit entities which are “related to” or “affiliated with” institutions of higher education;
Furthermore, the process by which a nonprofit is designated a "research nonprofit" is kind of arcane (for example, it's independent of how the IRS classifies the org in their 501(c)(3) designation). If the org you're applying to hasn't successfully sponsored cap-exempt H-1Bs in the past, expect additional delays while their lawyers argue with USCIS about it.
Those who want to immigrate to the US may also want to note down the green card lottery (aka Diversity Immigrant Visa) in their calendars. It is open to applications from May 7 - Sep 30. The basic facts about it are:
Whereas an H1B visa allows you to stay in the US to work at one employer that uses your specialised skills (and gives you a bit of time if you need to switch), a green card lets you stay in the US indefinitely, with any employer, or none at all. It requires you to have completed high-school, but not to have any university degree or job offer.
About 0.5%-1% of applicants are issued a visa (per the table below).
Countries with lower historical rates of immigration to the US have better chances, including Australia, New Zealand, and parts of Eastern Europe.
Citizens of some countries are ineligible: Bangladesh, Brazil, Canada China, Colombia, Dominican Republic, El Salvador, Haiti, Honduras, India, Jamaica, Mexico, Nigeria, Pakistan, Philippines, South Korea, UK, Venezuela and Vietnam (as of 2022).
The application is free. (Some companies try to scam you into paying a fee; don't!)
If you win the lottery, you can bring your spouse (and dependents), who can also get green cards. You and your spouse can both apply each year, to ~double your chances.
You can reapply each year. The window for applying and finding your results tend to be similar (May-Sep and April), so you can set the calender event to recur annually.
How much time does it take to get the paperwork in and apply for this for the 1st time by an org? You mention 1 week if it's an experienced org for whom this ain't their 1st rodeo - can you elaborate which parts take the time there?
The paperwork required to be entered into the lottery is almost trivial - see the step-by-step instructions here. Most orgs will want an immigration lawyer to do it though because while it's an easy first step, it's an easy first step in a long and difficult process. If an org isn't used to handling H-1B cases, I expect the huge hangup will be finding and retaining an immigration lawyer in the first place.