You mention TCS+, and as Aryeh's answer suggests talks and seminars may be a much easier and accessible way to learn about other results.
From my point of view (inherently subjective, and not too experienced, so take with a grain of salt) reading papers outside your subarea/subfield may be a good idea*, but going to as many talks and seminars as possible is a necessity. Below are a few things I'd suggest; in short, "going to talks, and absorbing information by reading abstracts."
In your institution, subscribe to as many mailing lists as you can -- combinatorics, machine learning, theory, whatever your department or adjacent departments (math, electrical engineering for instance) have. You won't go to all talks, of course, but at least you'll receive the announcements, and specifically the titles and abstracts. (This sounds somewhat silly, but this is important -- reading abstracts and succinctly describing what is new and what is going on in the other fields.)
Speaking of which, subscribe to arXiv notifications for several tags, or follow the TCS Blog aggregator and blogs like the Property Testing Review Blog. You'll get notifications about new papers, and sometimes a bit more info from a blog. (Disclosure: I am one of the writers for the PTR review blog.) Again, just reading abstracts and short blurbs about the new papers doesn't seem much, but it helps -- and if a paper actually sounds relevamnt, you can read it in full.
when in conferences, spend some time going to "random" talks not really applicable to your research at first glance. It depends how much time you have, and how overwhelming conferences are to you (they can be pretty tiring), but attending a few of these can be helpful.
You mentioned TCS+; this is indeed a good idea (full disclosure: I am one of the TCS+ organizers). Keep track of those talks, look at the slides, etc. There are other online seminars or recorded talks you may want to browse: e.g., the Shannon Channel, the Simons Institute videos (from their programs and workshops), the videos from the IAS lectures...
$^*$ if only to learn from the style and exposition of great papers.