Duplex Dweller

Check QRZ and look up LA1BRA. He's a Texan that works oil rig sites.
He has the same situation and uses a screwdriver. If you're only using one band or two, get a manual one. They work on 11 meters very well. If you look up his pictures on QRZ you'll see his setup and I've talked to him as easy as others in that area with small beams.

If you're running on a single band, A dipole would be better than a screwdriver.
Screwdriver requires a counterpoise, some big metal sheet. Cookiesheet?
Or just get a 102"whip and mount it on some piece of metal like that if your only on 11/10m.

A magnetic loop is probably the best antenna for stealth and multiband operation.
But expensive, just like the screwdriver.

You'll have to decide if it's working "good enough" or not. What you've got is a vertically polarized single 'leg' groundplane antenna. It isn't going to be the best by far, but all things considered, it'll 'work'.
Would it 'work' better if that vertical leg was pointed up instead of down? I would think so, but that doesn't seem to fit your circumstances, does it. How about adding a few more 'radials' to it (that braided leg)? It would certainly change it's radiation/reception pattern and could probably mean hearing more. If it isn't a huge hassle, why not? No significant improvement? So change it back.
Antennas would rather be away from stuff, above stuff rather than beside stuff, and generally don't 'like' being under stuff. Depending on your circumstances, that isn't always what you can manage. So, 'coochy-coo' them a little, get them 'close' to 'happy' and live with the results. That's usually called 'tuning', and it works.
- 'Doc

You want to DX; yet you dictate using a highly compromised antenna restrictions and mountings and expect results?!?
Let me tell ya right now: that dog ain't gonna hunt!

Do you mean DX as in the US or out of the US or outside of the US? To me, real DX isn't talking within the confines of the contiguous US of A (with the exceptions of AK, HI, and PR). More like talking to South America, Europe, Asia, Africa, Australia, etc.

Now that we got that out of the way, for a real DX antenna you might want to make and mount a horizontal dipole at/around 36 ft high off of the ground just to get a low enough "angle of radiation" when transmitting. Building it out of electrical conduit built in an "inverted V" form. That would be my choice; thta should work. You can build it yourself for about $20 - not including the cost of coax or the ~36 ft of mounting poles and dacron guy rope (not wire) you will need.

Now, that should work some real DX . . .

I think you're in the same boat most of us are in, making do with what'cha got. I wish I could do the "ideal" thingys with antennas but in most instances I can't, and never have. Oh well, do what you can and don't waste all that good 'worry' on it.
- 'Doc