We need more information. Where in the world do you live? A recommendation for a palm tree won’t do much good if you live in Maine. Is the fence you pictured yours? You mention a new taller fence - is that yours too? The pool to the right - is that yours? Is this picture taken from your property, looking at the neighbor’s house on the other side of the fence?

Hi! Sorry - I thought I initially selected Midwest Gardening as the board where I was posting! I’m in northern Indiana ( so far north I’m almost in Michigan). We are zone 5. This photo is taken from my yard , looking toward my neighbor’s house. You can see my fence( which will be replaced soon) and my neighbor’s house and deck . I’m the one whi owns the pool. Thanks, again!

Privet hedges might work if you don't mind pruning them to keep them under contol

Though it prefers sun and looks better when grown in it, one of the the most shade tolerant evergreens is Yew. Hick's would be the upright one you want. About 4' apart, center to center. (Could stretch a little more if needed.) Also look into 'Fine Line' buckthorn, which is claimed to be not invasive like the species.

Wow! Thank you for your suggestions!

There's another thread showing the buckthorn cultivar with a number of fruits on it. And it - like the rampantly growing virginia creeper - is deciduous = seeing through it all winter.

In general the quality hedges for shade are yew, holly and box. With different specific kinds of these having differing regional suitability. But attractive dense hedges like these take many years to get much height. Whereas fast growing kinds like privet tend to have a more sloppy appearance associated with their fast growth, unless kept formally sheared - as always in life there are tradeoffs involved with this subject area.

And with you being in USDA 5 long term cold hardiness will be an important factor.

Otherwise with that pine on your side of the fence and the other quite close to it on the other side being where the existing hedge attempt ends their presence is probably what killed the arborvitaes. (Although it looks like you planted the now highly prevalent 'Smaragd' - partial or complete failures both with recent as well as older hedge plantings of this particular variety are rather frequent, due probably to an assortment of circumstances).

Red margin bamboo.

From Wilson bros garden website: "Red Margin Bamboo is salt spray tolerant, grows well in sun or even deep shade, and tolerates poorly-drained wet sites. It provides a medium to tall privacy screen depending on your location."

In zone 5 it will grow to a height of 12' to 18'.

https://www.wilsonbrosgardens.com/Red-Margin-Bamboo-Phyllostachys-Rubromarginata.html

I think it would look good in front of the fence.

If you prefer evergreens, and can afford it, I would recommend searching hard and finding some large holly or yews. Large specimens will be expensive, but waiting 10 years for the actual privacy isn't fun.

One deciduous plant I had installed last year is red twig 'baileyi' dogwood. I have two of them in fairly shady spots and they are growing very nicely at around 6' tall this year. They look nice in the winter, too.

Bamboo of course needs lots of moisture to develop, if the pines were too much for the 'Smaragd' bamboo is not going to be a success under them either. Unless you install irrigation or hand water enough to get it established, keep it going.