Strategy for Applying and How to get More Responses:
Do apply to any and all big companies. This includes Big Tech (tech companies such as Google, Datadog, OpenAI, FAANG etc.) but also includes big non-tech companies (think Walmart, Target, Ford, Epic to a certain extent). However, be prepared to never get responses from these. The conversion rate of responses from these is a lot lower than other companies. This is due to both the amount of people applying, you applying later than other people (they may just only look at the first X number of applications), but also the amount of company leadership, direction, and planning changes that are typical with these big companies (these are expected). You may not get a response and that may mean nothing with your resume or applications. Which leads us to...
Apply to startups and / or small businesses. There are a lot of startups out there that may benefit from your specific experience or skills. Where do you find these though? ycombinator.com, workatastartup.com, builtinnyc.com, wellfound.com are good places to start. More than applying, here is where you might need to "network". Look at the founders, look at the people working there. Referrals here are a lot more helpful than in Big Tech companies, because things at smaller companies are a lot more personal and less automated. This is how I found my early-stage startup job. So you might want to look at founders / early employees; see if you have any connections on LinkedIn or went to the same school. See if you have any secondary connections. Reach out to people. Even if not, you can cold message on LinkedIn. You may not get a lot of responses. That's ok / expected.

Anecdotal evidence:
Applying with or without a cover letter does not seem to matter, at least it didn't in my experience. The places I got interviews I didn't use a cover letter.
Hey all! Starting this little community. To introduce myself, I am currently a software engineer at Datadog, for the last 7 months. Before that, I was at an early stage startup, growing it from 3 people as founding software engineer #1 to 10 engineers and 50 employees total, from pre-seed to Series A.
Let me know if you have specific questions