I am in the USA from the UK on business and have used the opportunity to take some holiday in the US before the work related part of the trip. There was a mix-up before I travelled and I was mistakenly told by an administrator in my company that I would be covered by the business's travel insurance for the whole trip, but now while already in the US I have been informed that in fact I am only covered for the part that is directly business. I didn't buy insurance before I travelled because of this and so now I am in the US without travel insurance. Can I still purchase travel insurance now I have already begun the trip?
A couple from the UK have a blog post dedicated to travel insurance for digital nomads. They share some tips and make some recommendations (note they are affiliate links).
It was difficult to find travel insurance policies that can be bought when you are already travelling but we did manage to find a few options and now always make sure we’re insured.
So yes it is possible to get travel insurance while already travelling, it may cost more and offer different levels of cover. But shop around to see what level of cover suits you.
I suggest you perform a search for "travel insurance while abroad" to get a list of possible providers. Then you can compare coverage and prices. Also search for reviews to see if anything negative appears (for example blog posts of travellers having a bad experience with them).
I had good luck with Allianz Global but... buying insurance after you already started on your trip? I don't think that is going to fly. Kind of like telling the roulette croupier, "Oh, yeah, that was the number I meant to put my money on."
Call them and ask.
I bought a policy from Globelink when I was already abroad and realized I forgot to take out insurance. They used to provide this option only via phone calls but this year they announced it can be bought as any other policy online on their website.