What a lovely garden!
As you've discovered, there's no easy answer to this - by 'removing spring foliage', does that mean you've got lots of bulbs in under the trees? Because what I'm about to suggest won't work if you have - I'd use a single evergreen shrub beneath, chopping out some root to plant it, and then let it grow to fill in the space year round, something like Sarcococca or Prunus Otto Luyken. Otherwise, if you do have spring bulbs, you need something that isn't there in spring, but is in summer - like a fuchsia, but the trouble with that is, you'd need to make sure it was kept well watered during the summer.

I use hardy geraniums a lot in those sorts of places. The macrorrhizum group of geraniums seem to thrive no matter where they are planted.

Thanks Bamboo, I like both of your suggestions but I do have bulbs in spring.
I have some stray geraniums growing in another border which I planted in the first picture. Keeping fingers crossed that it will take.
I do get frustrated as it seems to spoil the borders every year and I don't seems to get it right.
Thanks to you both

the other thing you could try just for ground cover is Lamium maculatum, maybe White Nancy or Beacon Silver. They spread, and tolerate dry conditions, but the bulbs should be able to come up through that anyway. You can cut back and pull out bits where necessary if it gets too big.

I've had a White Nancy for a couple of years and had to move it recently as it just wasn't thriving.

I wondered about a fern - some are OK in dry areas, dryopteris felis -mas for one. it gets quite big in summer and it does disappear in winter. I was going to say what Inverglen suggested too.

I have Lamium orvala, it grows to about 30" has much larger flowers on than the smaller varieties, is much loved by bees etc and likes all types of shady situations.

That's an interesting one, 2ndhand, new to me, just checked it out...

Thanks 2ndhand. That is nice. I do infact have lamium purpeum and took seeds last year and have a lot of smaller plants. Do you think they will be ok under the tree also ? Like the idea if ferns also .

Don't see a problem under a tree. infact a couple of nice ferns behind would give a nice contrast on leaf shapes if you go for the smaller Lamium.