Need Advice on Lateraling

I currently work at a balance sheet bank (think Truist, BMO, etc...) and am looking to lateral to a BB/EB. Recently, I have been actively sending out networking emails, but I have been struggling to get any responses. I graduated from a non-target school and had much more success during my sophomore internship networking compared to my lateral networking efforts now. I think my resume is solid despite not coming from the most prestigious school/bank, but I thought I would have gotten a couple of bites by now (if I am underestimating the difficulty of lateral recruiting compared to internship recruiting please let me know).

For people that have successfully lateraled, did you find more success cold emailing for roles where there was an active job posting? Or does a job posting create so much demand that my networking efforts will get lost in people's inboxes? If you did cold email where there was no job posting, how did you convince people to get on the phone without an opening in their group?

Thanks in advance!