Hello from JustAnswer.
Your doctor implemented the correct interventions
for acute management of a jammed finger. The key interventions for the first couple days is RICE - rest, ice, compression, and elevation. Ice is better to be one for frequent, short periods of time, rather than prolonged use that can cause cold injury. A splint and wrapping is good compression for a finger. Elevation should be above the level of the heart. Steroids or non-steroidal anti-inflammatory medicines may help by relieving secondary inflammation. You do not say what strength of Motrin was prescribed, but the dose of Motrin can be adjusted up to a maximum dose of 2400 mg per day. If you are taking the 800 mg strength, then you are already at this dose, but if you are taking a lower strength, the dose can be adjusted.
After 48-72 hours, it will also help to apply moist heat, either with a moist heating pad or hot compresses.
The reality, though, is that the primary need at this point is time. A jammed finger involves injury to fibrous tissue, specifically the tendons and ligaments of the finger. These structures will heal slowly because of the nature of this tissue. The interventions above will provide the best environment to support healing, but it will still take some time. If you were in a critical game, such as a professional athlete in a championship game, then it may be an option to inject numbing medicine and tape the finger to the adjoining finger. But for most situations, it simply will require time to heal.
If I can provide any further information, please let me know.