Ideally, a trellis should run north-south so as to shade as little as possible.

So if your trellis is on the east side, running on the long edge north to south then you will probably be shading the least possible amount of garden that way.

Do you have high winds?
I put my frames for peas in an A shape running E-W. Everything was great, but we had wind last week and now my pea plants have all been blown the the west end.
Next time, I will align my frames N-S so that the prevailing westerlies will blow my peas ONTO the frame rather than OFF OF or ALONG the frames.

More to your point, if you are looking to align true south it is probably ok for your latitude. The higher your latitude, and the longer/higher your trellis, the greater the shading effect on your north end. Your east will get morning sun only, which is another important point.

Good luck.

Ideally, the trellis and any taller plants should be on the north side (running from east-to-west). The sun in these parts is in the southern sky.

Not always possible but that's the reason.

Haha. Actually the explanation leaves some uncertainty.
You do not want to change the orientation or size of the plot, of course. You say there is a rail fence. You care about what is on the other side (east)?? If you do not care, then that is a good place for your trellis. It will shade your garden in the morning. If the trellis is true N-S, you will lose about half of your insolation by putting a trellis there.

If you put an E-W trellis on the north edge of your garden, you will shade whatever is north of that by quite a lot. A triangular area behind it will get no sun.

Consider putting half length trellises running N-S on the north part of your garden, which will give you some shaded areas, but the southern part of your garden will get full sun.

Consider also that if your trellis is mostly translucent, it will not shade things much until the plants are actually growing on it. So it will give shade in hot summer, but let sunlight in during other times. Maybe that is good.

I do not want to belabor the point, but let me just say that I misjudged true south in my garden by about 30 degrees for the first two years. Who cares? No big deal, right? No. Huge difference. Plants that I thought were on the east end of my garden were actually northeast. Basically my kitchen is in the northeast part of my house, and I wanted a garden near the kitchen. OOOOPS.

I run my bean trellises N-S along the long side of my 4'x12' beds and have had good results growing tomatoes and peppers alongside them. The sun is sort of a non-issue because by the time the beans reach the top of the trellis, the other plants are all good size and are busy fruiting. I think a little shade is welcome at that point to keep the soil a little cooler. I am also in Zone 6, and we have hot, dry summers here. I use jute twine or bird netting on a frame, so the trellis itself blocks almost no sun when there aren't any plants on it.

I also think I should mention, that pole beans will want a minimum of 6' for a trellis. Mine are about 6.5'-7', and the pole beans always top the trellis and turn around and come back down. 5' should be fine for Cukes