How Do I Negotiate Higher Salary?

I interviewed for a role and feel extremely confident that I will get an offer by the end of the week, but- I know I low balled myself on the salary. Over the phone I asked for the pay rate, the interviewer asked my salary range, and I said $42k-$45k. She said they offer way more than that for the role and I proceeded to schedule my in person interview. When I went in for the interview everything went really well, she showed me my office, introduced me to my team, and gave me a tour of the facility. We got to the salary portion and she asked if $55k would work for me and I said yes. In further discussion she mentioned that since I am starting this late in the year that my first raise would not be until Jan 2025. In my last role my salary was $39,900, so this new rate of $55k really is a boost for me, but considering the wait for a 3% raise, I’m thinking that I should have gone higher with my salary request. When I got home I had an interview invite from the companies competitor and scheduled for Thursday. I started thinking about the amount of work I will be responsible for, and the dollar amount I would need to live comfortably for the next year and a half. I realized that with my experience level and qualifications that a reasonable rate for the job is $62,500. Upon receiving my offer letter, how should I go about negotiating this increase?

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