Hi Hilleyja,
Reading through your post a few times, I half assed my way through drawing up a diagram of what I think your situation looks like.
The first thing that strikes me as odd is that your connections vary so much between day and evening. That said, there are a few things that you might try to up your signal strength before spending money on this TRENDnet adapter. You might try a different channel. I believe there are 11 different frequencies that you can set your wireless G to operate on and simply changing them might help out some. It doesn't take long to try and it's free.
I have never worked with that Linksys range extender so I am unsure how they perform. I am however questioning it's power if your still getting poor signal strength even with line of sight. While I have worked with the 637AP extensively, I cannot guarantee that it will solve your problems. The adapter needs to be wired to your router so that alone will limit your placement of the thing. You can run a length of cable if you want to place it someplace different but having Cat5 running across your floor, or under your trim (however you want to run it) is clearly a choice you will have to make.
That said, if the PC in your outbuilding is operating on the G band, you still might encounter the same problems your experiencing right now. The range of N is a lot better than G so if you want to truly take advantage of this new protocol, you'll have to buy a N adapter for your PC in the other building and thats going to add cost to your project.
The TEW-637AP is a great little device and has enabled me to expand my signal all the way out to my barn with is about 60 feet from the house. Like you, I have the adapter setup close to a window so wall interference is kept to a minimum.
I can't guarantee that this device will help you out but I personally think that it will. I am basing this solely off of the information that you gave in your previous post too. If you need any additional help, please stop by again and ask. I will work with you to get a respectable signal in your other building.
*EDIT - You also will not have to change your network settings on your existing router to use this AP. Simply hook it up and run the setup utility. Remember though, you will see the TRENDnet's wireless SSID and not your routers, even though you will still be on the same network.