Sounds like you may need to move the hydrangea instead, if they are as close to each other as I pictured them to be. The dogwood's shallow roots may compete with the hydrangea's roots.

If it is unwanted, I would be tempted to prune it down and keep cutting new growth until it stops putting new growth. But due to the shallow roots, extracting it would disturb the hydrangea so I would not try that route. That is what I do with Crape Myrtles that grow as weeds here.

If indeed a very *large* hydrangea, I'd recommend digging up the portion of the root ball affected by the tree, and then dealing with the dogwood's main roots and stump once exposed. You can hack them out with a hatchet or use a hand saw. Hydrangea roots are very small and dense (like hair) so you can easily move and pull clumps of them aside. The tree roots are much less numerous and thicker.

If the tree is smack in the middle of the crown of your plant, you might even have to split a portion of the crown in two to get the tree out, but a large plant should survive with the crown in two pieces, as long as roots are attached to each side.

This could even give an opportunity to rework or even improve the soil. Just bury the clumps of roots in some rich soil and pack back down.

This would be about the cut-off time of year to do this. Any later, and you'd worry about dryness or the heat fatiguing the plant directly after replanting. Either way, now would be the time to take action on the tree. Lesser of two evils.

The dogwood is actually at the edge of the crown - maybe a little inside it. What do you think about tying it to the chain link fence and putting a plastic bag over the leaves? I have done this with another volunteer tree - continually strip off the leaves and while it hasn't killed it it has kept it from growing much larger. There is just a long stem in that case that I keep hidden under the plant. It is much thinner and more bendable though.

If you "do not want it there", I would cut it just cut it off less than 1" from the ground to get it out of there faster.

Originally I was going to say that you could dig a 2-3" small trench towards the dogwood and hydrangea until you got to the dogwood. When you get to the dogwood or close, then cut it off and prune the roots going out of the main dogwood branch to see if main branch of the dogwood will "disconnect" itself but that may be too much work.

I realized I didn't make it clear but I do not want the dogwood there. So, Luis, if I cut it off right above the ground, you don't think it will sprout leaves and branches there and be a worse problem?

Yes but if you regularly cut the new growth, it eventually will stop. That is what I do with my CMs. I have a few that grow under the radar and when I finally get to them, it takes 1-2 seasons to make them stop.