Ginger and lime, one of the best flavor combinations ever? I would say so. The southern baptist, which I found while browsing the interestingly named SquirrelFarts drink blog, combines both ingredients beautifully.
Although rye seems like an unlikely spirit to use as the base for this one, the spiciness of the rye combines well with the ginger warmth and because of this I reccommend using a rye with high rye content, such as Bulleit. The lime then acts to calm down the other flavors while adding a nice tartness and the sugar in the ginger syrup smooths it all out.
For the ginger syrup most recipes call to muddle fresh ginger with simple syrup but I took a different approach here. Since I already had my homemade ginger tincture (mentioned previously in Old Rye Gal), I decided to just make a syrup by combining the tincture with agave and water in a 1:1:1 ratio. Depending on how strong your ginger tincture is I would consider modifying the ratio accordingly. The fresh ginger method also seems like it would work great, and would probably add a little more of the sharp spiciness.
2 oz Rye
1 oz Fresh Lime Juice
3/4 oz Ginger Syrup
Mix all ingredients and shake on ice. A small piece of lime zest works great as a garnish on this one.