Yes, you can shift into a generalist role. Try to avoid offers for recruitment and get into any company with any role apart from recruitment, which will help you to grow into a generalist profile. Maybe in the first couple of years, you might be given specific assignments, but if you perform well, you can certainly achieve your goal.
Basic Skills for a Generalist Role
The basic skills I would look for in a fresher are communication, presentation, interpersonal skills, and good subject knowledge. In any interview, three core things will be assessed: skills, knowledge, and attitude. If you are positive in these three areas, your path to becoming a generalist is just ahead!