Hi New WV Carver!
You can't go wrong by adding a dog leg chisel to your tool kit. The tool is a delight for those of us that use the undercut technique in our relief carvings.
I have both a small micro dog leg chisel made by dockside and a large one by Ramelson ... both are very reasonably prices beginner tools.
A wide sweep gouge will also do a great job for you ... and as you are a new carver just filling their tool kit ... this is the one that I would suggest. Unlike a round gouge that is usually has a half circle profile the wide sweep has only a very slight curve to it's profile. So it is one small arch of the circle giving just a nice little curve to the edge.
You can slide the edge of the wide sweep under the upper cut of the undercut to make your second low cut. The sweep gouge is wide enough to make very deep cuts under your tree branches or fence rails.
Now the reason I am suggesting this one for you is that as a new carver you want to add tools that you will use with every carving ... not just undercuts. The wide sweep is great for leveling your background area and removing the ridges left from round gouges or the scratches that can be caused by chisels. I use mine to smooth the surfaces of my main elements when I want to remove the chisel planes.
So I think you will find the wide sweep a double pleasure.
The detail knife is an excellent choice. But please know that this knife is not meant to every be used as a miniture pry bar

It's thin edge is meant for cutting strokes only and I would not suggest that you use it for undercuts as you can easily break the point.
If most of your tools are too bulky I would put a mid-prices beginner's kit of micro tools on your wish list. Such kits usually have five or six different tools including chisels, skews, gouges and v-gouges. They will last quite a while, get you started using some smaller tools. And when you know which tool profile that you prefer you can add a higher quality tool of that profile to your kit.
Hope this helps.
Susan Irish