I'd look into a dual-degree program (if you attend an LS with a school of public health you might be able to create your own...) or I'd plan to complete the MPH after LS. I would strongly recommend against an online degree - you would miss out on the chance to connect with faculty members and fellow students, and the five-year timeline needed to complete the online degree is NOT worth it. With a dual-degree you could probably finish LS and your MPH in 4 years and be competitive for the job you want straight out of LS.