Sorry for the delay in responding! Unfortunately it did not work out with my US company because we couldn’t agree terms, so I am not currently working for them (or anyone - still searching!). When I was researching, it seemed that going through an umbrella company was going to be the most straightforward and “safe” way for me, but I am sorry to say I did not get much farther than that.
I would also just caution you to be sure to agree favourable (to you) terms for a contract. I know we all want to work and any income is good income in this phase, but I personally experienced a situation where my US company felt they could take advantage of that (just the tip of the iceberg was feeling that they could pay me far less). It was clear that the transition from employee to contractor wasn’t going to be a happy one, and I wasn’t keen on spending my first months here working from home, alone, and feeling miserable and taken advantage of by a job thousands of miles away - all while adjusting to everything. In the end, though I’m still job hunting, I am 100% sure that it was the right decision for me to walk away. I know it is a luxury to be able to say that, and I am sure you must work for nicer people that that, but just wanted to share my experience as a caution.
Keep us posted on what you do!