Lateral to a larger PE shop

I am an analyst at a middle market investment bank (think Harris Williams / Raymond James / William Blair etc.) on the West Coast and will be joining a lower middle market private equity firm in the midwest as an associate. I went to a top 15 undergrad school but not having a BB IB on my resume made it very difficult for me to even get interviews from larger middle market firms and megafunds. I intend on spending a year at the lower middle market PE shop and then making a lateral move to a either an upper MM or a megafund in NY (ideally). Would love to hear from someone that has gone through the process before and learn how difficult or even feasible this process really is? I am also open to going back to a tier 1 investment bank as an associate (if it's still possible) or business school if these options would make it easier for me to land a gig at a megafund.