TL;DR: Start job hunting, as the process takes time. You can always negotiate the start date if needed.
According to this page citing a GlassDoor study on the duration of the hiring process:
It takes 22.9 days for a candidate to progress through the hiring process in the United States
Additionally the article provides higher durations for larger businesses hiring for more specialised roles, e.g.:
The length of the hiring process also differs by position. Jobs with the longest process were typically government, academic or senior-executive positions. In the United States, police officers report the longest process, at 127.6 days, followed by patent examiners, at 87.6 days; assistant professors, at 58.7 days; senior vice presidents, at 55.5 days; and program analysts, at 51.8 days.
Lets say that the hiring process takes 60 days seeing as you are interested in working at larger, more bureaucratic companies in presumably a specialist role.
Moreover, we have to factor in the time it takes to job hunt and apply. According to this page:
Recruiting software company Jobvite crunched the numbers to find out exactly how long you can expect to be in job-hunting limbo. Overall, the average job-search process takes just over six weeks— 43 days, to be exact— but that varies considerably by industry.
So it could take about 100 days to land the job, i.e. just over 3 months. Given we are talking about averages, then your mileage may vary.
One final point is that when you land your new job, the start date is negotiable. If you foresee that starting early clashes with your academic responsibilities, then explain this to your new employer and negotiate the start date.