Ten watts is plenty, that wire is your problem. A foot off a chain link fence is basically an antenna glued to a big lump of metal, and the garage siding isn't doing you any favors either. All that stuff sucks up your signal and turns it into heat instead of radiation. Same wire up at 20 or 25 feet, clear of everything, would sound like a completely different station. I ran a similar end fed slung over a roof ridge for years and the difference between that and a low fence run was night and day.
The night to night swings you're seeing are probably a mix of both things. Band conditions do change, especially in the evening when 40 meters is transitioning, but a compromised antenna exaggerates that. When you're already down 15 or 20 dB from losses, a modest shift in propagation is enough to push you from barely heard to invisible. Stations with efficient antennas ride through those swings without noticing much.
First thing I'd do is check the match. The little matching unit that came with the radio will often load up a bad antenna and show you an acceptable reading, which fools you into thinking everything's fine while half your power cooks the matching unit or the fence. If the match changes a lot when you touch the wire or move around the yard, that's a red flag for losses in the system. Try this: run the wire higher and as far from the fence and garage as your yard allows, even if it's shorter. A half size wire up high beats a full size wire in the weeds every time. Add a proper counterpoise or radial wire on the ground if the matching unit needs one. Then spend a few evenings comparing, ideally working the same stations around the same time each night so you get a fair test.
And don't give up on the low power. I've worked plenty of DX on 5 to 10 watts, but always from an antenna that was actually in the clear. QRP punishes antenna problems hard and rewards good ones, which is honestly the best part of running it.