Nearly 2 decades ago, I started out as a help desk (tier 1), after 2 years with one org took an outside job with better pay as a Sr Help Desk Support. Around 3 years later, I was promoted to “Jr systems and network admin” and I no longer had to deal with end user support directly, and became more of an escalation resource with ongoing admin responsibilities.

I assume this is what you mean by sysadmin, where you might be in charge of AD domain controllers, domain management, network configurations, backups and disaster recovery, etc. and not helping a user fix a printer, unless the issue has been traced back to one of your responsibilities.

In my opinion, it is easier to get that first bump to tier 2 within an organization. I’ve never seen an external hire with strictly “help desk” or end user support transition into a sysadmin role. It’s always been someone internal moving up, or hired with specific experience if it’s an external hire (they already have a couple years of tier 2/3 experience).

Pick some useful tech certs to study, you don’t even need the test for the cert unless a job mandates it, but you will learn useful things and discover personal interests that you can integrate with career goals / progress.

Identify something your team underutilizes, something that you can take growing responsibility for and learn how that thing works better that thing works better than anyone you work with. Bring useful improvements to your team and you will be seen as a problem solver and opportunities will present themselves.

Just know that IT is a constantly changing landscape, and learning a technology that your org uses, even if it seems outdated will give you a firm foundation upon which to grow.