For that type of injury, topically is the way to go. Get a container if you can find one that's just a bit larger than your finger.
Whenever you don't have time that allows you to relax like watching TV, or any thing that doesn't require two hands soak the finger in CS. Or buy a small spray bottle and spray the finger whenever possible.
The more times you do either way, the better.
CS will kill both the good, and bad bacteria in the gut.
Chlorine Dioxide Solution is most commonly called CDS. The easiest way to get it is to make it yourself with MMS and Citric Acid.
The solution(when sold prepared), comes in various concentrations, though I wouldn't get anything under 5000ppm as CDS looses it's strength over the course of time. I'd also look for a solution that has a packaging or bottling date on it so that you can determine how fresh it is. Otherwise, you'd be far better making it yourself.
You can buy Sodium Chlorite(MMS) from eBay: http://www.ebay.ca/itm/20-grams-Sodium-Chlorite-80-Powder-Flakes-make-your-ow...
You can buy Citric Acid pretty much anywhere(bulk stores, canning store, wine supply store etc).
You can then make CDS(quite easilly) using a process called the overnight or shot glass method:
Step 1: www.youtube.com/watch
Step 2: www.youtube.com/watch
In conclusion, I'd add that CDS and DMSO is perhaps one of the most powerful and rapid use topical antioxidants available. With uses ranging from Staph infections through to gangrene.
Some people drink CDS and or other MMS variants, though I wouldn't recommend such measures for all but the most extreme of circumstances(life and death).
Hope this helps