According to https://lawschooli.com/best-law-schools-for-biglaw/, large (500+ lawyer) firms pay 3x what small firms do. So the competition for those jobs is going to be tough unless you graduated from a top law school. The article says that 88% of University of Chicago graduates went to a big law firm in 2017 vs 4% from Texas Tech.
However, a lot of law graduates do not go to big law firms because they don't want to. There are lots of different ways to be a lawyer, and practicing at a big law firm is only one of them. Many law graduates prefer to work in midsize or small law firms, where they can often get more hands-on experience early on in their careers. Also, big law firms don't tend to do certain types of law, like family law, criminal defense (although they sometimes do "white collar defense" for certain types of fraud and financial crimes), immigrations, etc. Other law graduates want to work in public interest, which is a huge category that can span direct legal services to people who can't afford to pay, to lawsuits and policy work for a particular issue (civil rights, environmental protections, racial equity, etc.). Still others go to work for the government as prosecutors, public defenders, legal analysts for Congress, and as lawyers in every agency in the federal and state governments.
After I graduated law school, I went to a big law firm for a short period of time. I got the job through the normal process: The law firm came to campus and participated in our on-campus interviewing program. I got a job working for them the summer between my second and third year of law school, and at the end of the summer they gave me an offer to join the firm when I graduated. I found out it wasn't for me and moved on to the government, and eventually working in-house for a corporation. I do think that having the law firm on my resume helped me with getting those jobs, but it wasn't necessary -- people also get those jobs through other routes.