What is your definition of BIG4 Strategy Consulting? If you mean Strategy&, EY-Parthenon and Monitor Deloitte. The internal networking possibilities will not help you. Therefore I would recommend joining BDO Strategy and then try to apply to them, depending on your profile you will need special expertise or an MSc or MBA from a target university.
If you define BIG 4 Strategy consulting as a team within the BIG4. I would recommend doing Audit withing the company and then network aggressively your way to the strategy role. Specifically, you can ask partners for coffee, most of the time, they will say yes.
Hope that helps. If you have more questions please let me know, I'm always glad to help.