The most effective and safest solution would be to call a tree service. They will have the equipment to reach and cover the tree in its entirety which is essential for proper control.
Otherwise you need to get yourself higher (ladder or get on the roof) or find a more powerful sprayer.
My Ortho Dial-N-Spray sprays farther than my pump sprayer but I don't think that will be high enough. I can get about 12-15' with it.
If you have a pressure washer you can pick up a an injector for it and that might be able to give you a little more reach. That still might not be enough but in combination with a ladder might do the trick.
Alternately, have you considered purchasing and releasing lady bugs as a biological aphid control? Sounds like they'd love that tree.
One more option I haven't used myself but claims to reach 30' high is a trombone sprayer. They cost around $60+.
Most of my direct experience with tree spraying was for adelgid but I had to look into the same types of dormant and horticultural oils as well as insecticidal soaps.
Your biggest issue seems to be not getting total coverage with the dormant oil. It sounds like the trombone sprayer will be the best option and costs less than a single visit from a tree service in my area.
Once the weather gets warmer, if you still have an issue you can use an insecticidal soap/horticultural oil or just regularly spray with a hose to knock the aphids off.
You said the honeydew gets on the car so I would think that would mean the tree is close to your home. If you can have someone throw a hose up at you while you're at an upstairs window (if you have one near the tree) you might be able to cover a decent amount. Either that or get on the roof or prop a ladder up near the garage wall. That's the cheapest, least toxic way to deal with it but you have to be careful.
In the past I used Volck Oil. Depending on the dilution it can be used as either a dormant or horticultural oil. The next year I still had some adelgid and I switched to this combination Neem/potassium fatty acid spray that is no longer made. It worked just as well as did plain neem oil but things weren't as bad by that point. I haven't sprayed in a few years. I just hose down the tree from time to time with water and haven't noticed a reinfestation.
The most important thing when spraying with a horticultural oil is to mind the weather and do a small test to make sure it won't damage the tree. I used to get up before the sun on a day that was supposed to be cloudy. A day when it's supposed to rain is great since the oil/soap kills instantly so it doesn't matter if it gets washed off after spraying. If you have an infestation and a heat wave I would just spray as much of the tree as possible with plain water in the early morning. At least over the car.