One of the possible approaches: Make the analysis of Your targeted audience that subscribed to Your Great Page or visit Your page regularly. After that, You will get a clue which potential sponsors might be interesting in Your offer (being Sponsor). You can reach them via Socla Media or just e-mailing them with offer of cooperation.
see if you want to get sponsors for facebook page with 1 million likes if
1.you have to do something unique in your page first he might get intersted about you then
2. if you get viral or become with less amount of time
3. or get a contact with head manger personal account to talk about sponosors
4. the best option you can call me because my charges are affordable than others and may get a new ideas who know
Quite a complex question, but I could provide you with an answer and an initial solution. I have recently created my book "answer" that addresses your problem, so if you are interested you can order a free call.