Get experience in any public firm. The closer the client types are to the city/client base of some of the Big 4 the better. GT/Mcgladrey/BDO are going to be somewhat similar in terms of what you'll learn from an audit standpoint.
Your best bet in switching into Big 4 is with 1-3 yrs of public experience. The most reliable way is to form a relationship, or at least a working/professional acquaintance, who would be willing and able to vouch that you are competent and pass your resume to recruiting.
Once you do that, then you just have to find the right timing. Most firms have regular needs for experienced hires, but your timing and availability will have to match up with an opening that comes up. Your best bet is to form that relationship/acquaintance and keep in touch until the office has hiring needs and then go for it. Repeat with as many firms as possible.
Also, having your CPA helps a lot since it alleviates a lot of the fear (justified or not is a different point) that you are competent.
Lastly, don't close yourself off to enjoying a mid-tier firm! Some people will enjoy good career opportunities both inside and outside the firm. I wouldn't ever discourage someone from aiming for the Big 4, but make sure you give a fair look at all opportunities. Even outside of public, in industry, there are some analyst programs that can be very good for your career.
Good luck!