It seems, old secondary sources (1920-1970s) are widely cited in modern works and often provide exceptionally detailed view on the topic. Like a dozens-fold magnification of a citing paragraph. The danger stems from their age. Hence the questions:
This is going to depend heavily on your field, and also the specific subject. I study the New Testament and early Christianity. If someone cited Wrede (1901) or Bultmann (1950s/60s) without any reference to more recent work, I'd be very skeptical, but for topics related to those specific authors, "more recent" could easily be the 1980s or 90s. If someone cited Conzelmann's commentary on 1 Corinthians (1975 in English, I think earlier in German), depending on what they were trying to do I might not consider the source outdated. But those are all regarding the biblical text itself. If I were reading something about social issues in the Pauline assemblies I would expect a big chunk of the major secondary sources cited to be less than twenty or even fifteen years old, and many things older than that would be included primarily as part of a "history of the study" section.
Basically, if the last time someone wrote a comprehensive or worthwhile treatment of your topic was 1965, then you're going to cite a book from 1965. You'd treat it with the same critical analysis you should use for any source, and you might end up flagging a lot of flaws, but it's still the most recent work. If it's so outdated as to be useless, and no one's replaced it, then your options are either to pick another topic or go back to the primary sources and start from scratch. Maybe you just found your thesis topic.
As to whether new data can invalidate older sources... again, it depends on your field. In many fields of pre-modern history, the data doesn't really change that much. It gets nuanced and fleshed out by archaeological research, but the further back in history you go, the more rare it is to find new data that completely invalidates older data. It's more common for approaches for the data to change, and for people to question or debunk earlier assumptions and therefore the conclusions they engendered. If your field is more modern history, or if the archaeological work for your region/period is more limited, it's possible that new data appears all the time, and so maybe it's the case that any secondary source more than ten years old is suspect.
Finally, how to find newer secondary sources. Step One: Make friends with your librarian. That's why we're here. We can show you all sorts of secrets for Finding the Things. Most library catalogs have ways to narrow and sort results by date. Google Scholar's "cited by" function, which you already mentioned, is helpful. Subject heading links in library catalogs can also be really helpful--they're a little bit like going to see what other books are on the shelf in the same area (which is also a helpful strategy if you have access to a physical library). Some fields have bibliographic indexes that are published quarterly or yearly, though those are less common now that we have the internet.
Varying your search terms in library catalogs or Google Scholar is hugely important: "Victorian fashion" will find different sources than "Victorian hats" or "haberdashery England 1890s." Another strategy is to look for works on related topics. A book on gender in 19th century England might have a chapter or two on clothing, or even just a short section with some thorough footnotes, but it might not have shown up in any of those other searches. I'm currently doing research for a possible book on spiritual practice in early Christianity, which is a topic with weirdly sparse scholarship. But a lot of works about Roman family structures, social formation, civic leadership, etc., have chapters or sections that are helpful. Some people recommend mind-mapping strategies as an early stage of research, to get on paper as much as possible that might be even tangentially connected to your topic.
Finally finally, if you have access to it, networking can be a life-saver. Maybe someone you know went to grad school with the person who unearthed a vast treasury of Victorian hat sketches in some archive. Maybe you go to a conference and make friends with the guy who is currently writing his dissertation on Edwardian hats, and he will talk to you about it for days. Scholars are stereotypically cave trolls who hiss at the sun (no hate, I count myself among that fine number), but scholarly communication is an inherently communal act. Your fellow historians are resources for you, just like the library catalog is.
On the question of resorting to old sources, I find the period 1920-1970 (and maybe through 1980 or so) to be a golden age of extremely high quality scholarship in English in the fields I was trained in: early modern world history, history of colonialism, history of slavery, and indigenous societies of the Americas. While there are many merits to more recent scholarship, and also an enormous value in exploring 19th century efforts as well, you will find well-known works of the 1920-1970 period to be of generally very high quality methodologically. On key points, follow their footnotes to their sources, and always be honest in your own citations (citing the secondary source when you cannot go to the primary yourself). You will find mistakes and manipulations in all generations of scholarship, but I find the half-century you mentioned, or perhaps 1930-1980, to be the strongest period of English-language historiography in the fields I actively work in.
A particularly valuable type of resource from that period is printed collections of archival primary source documents such as the Hakluyt Society volumes or the Slave Narratives collected by the WPA Federal Writers' Project. Even printed primary sources should be used carefully if you are publishing your own research as they can contain important errors, especially of transcription and translation, but you will find many high quality works from the period in question.
If you have university database access, I find Oxford Bibliographies an excellent resource for pointing me to up-to-date literature on a wide range of subjects in the social sciences. Unless you're on-site when accessing it, I think you'll want to find a link from your university library page, but I'll place a link below to try and be helpful.
Here you go: https://www.oxfordbibliographies.com/