I've been on Fentanyl for over 2 years for chronic pain that can not be surgically corrected. The first year, I didn't need any sort of extra for break-through pain; even the lowest dose was strong enough to keep me both pain free and mobile (if a little sleepy now and then). More recently, I've gotten morphine instant release added to the regimen, even after increasing the dose of the fentanyl a little (by accounting for metabolism and skin in how/where/when I wear a patch) it still doesn't provide the same amount of relief. Rather than double the dose of the main medication, which is unfortunately the next available step, I get an adjunct.
Not to disparage, but 4 or 5 months around a known accident is not chronic pain. Chronic pain would be if your hand continued to hurt (even a 5 out of 10 is considered "treat this") years later. Level of pain is only an issue in how much treatment a person needs, not the kind of treatment that's used. What sucks is that our options for treating low level chronic pain are tylenol (which causes liver failure in large doses), aspirin (causes gastric bleeds in some), and . . . that's about it. A step up you have tramadol and tordol, the first a synth-opioid that can't make you high but can raise serotonin levels so much that it can't be prescribed with most psychiatric medications (and if you have chronic pain and aren't depressed about it, you are rare) while the second is a very strong version of aspirin that causes even worse gastric bleeds if you are prone to them. Above that, opioids, just opioids. Well, there is burenorphine, a synth-not-quite-opioid that can be used for pain if you aren't too tolerant of the other drugs, but it's only got a small therapeutic index for pain (starting dose is 1/4th the max dose) and switching from the lowest fentanyl patch to butrans patches required the second strength level, so not a lot of room to go up.