You will need to get one x ray done.. There is a possibility of fracture or dislocation in or around thumb joint.
Next Steps
You can start ice fomentation, continue splint for support, take analgesic with anti inflammatory medicine.
You may require to get one x ray of hand to rule out bony injury or joint dislocation. Better to get it done as early as possible..
Health Tips
Keep the limb elevated, avoid movements of thumb..