1. Take the URL / link / address of the page the call is published on and stick it into Twitter etc to see what people are saying about the call and who is doing the saying.
When applying for a grant a few years ago this helped me discover an interview with two of the people involved in the grant we were applying for. They were being asked what they were looking for and promoting the scheme. This interview was incredibly helpful but was not particularly flagged up anywhere and I pretty much found it by accident.
Often, but not always, the first people to share the link will be involved in the project or organisation in some way so may have useful things to say about it.
2. See who has been funded previously and the type of project they did. I might search for the name of the grant on Twitter etc and add in words like "delighted" or "announce" etc.
Clicking on the link in their post or the website link on their profile can give me more info about that project but searching on Google can also put that information in context. This strategy recently pointed me towards an evaluation of a previous instance of the grant (because the organisation was listed in the evaluation), which also pointed me to more funded project examples and more info about what the funders had been looking for.
Commentary
I think of this background research also as 'chaining' or 'triangulation' - using one bit of information to find another, then using that to search for other stuff and so on. Possibly everyone is already doing this and so I'm not doing anything unusual, but I've not seen it explicitly mentioned (or failed to notice because I'm newer to grant applications*) so thought I'd write it down here in case it's useful.
Background
Although I've worked in academia since 2010 it's only more recently that I've been involved in applying for grants. I've previously applied (successfully) for local small grants up to £10k and been on a few decision-making grant panels at that sort of level of funding.
More recently I've helped on some larger grants and am currently waiting to hear the outcome of one and planning an application for another mid-range grant when it opens again next year.