Hi
Yes you can shift in to a generalist role, try to avoid offer for recrutiment and get in to any company with any role apart from recruitment which will help you to grow in to a genmealist profile. may be first couple of you might be given specific assignments but for you perform well, for sure you can achieve your goal. The base skills, i would look into a fresher is Communication, Presentation, Inter personal with the good subject knowledge.
Any interview is concern three core things will be assessed that is Skill, knowlegde and attitude. If you are positive in this three things. Yes your generalist is just ahead!!!!!