In the immediate aftermath of the Winslow Event, Taylor had some issues with power control, much like Vicky canonically / fanonically does. Danny's truck is probably a small truck, Ford Ranger / Chevy LUV / Dodge Dakota, probably late 90's/early 00's. Taylor had a problem folding herself into the cab, and didn't want to risk accidentally wrecking the thing trying to get in. He doesn't have the monsters I see every day - F150/250/350, Dodge RAM 1500, Nissan Titan, Toyota Tacoma, and the large Chevy trucks (I live in farm and cattle country) running around the city, unless someone from out in the countryside is coming into town.
Yeah, I know there will be someone in town who has a big pickup truck, and have reasons (good or bad) for having it. But Danny doesn't strike me as being one of those people.
Likewise, once at the house, in skipping over the broken step on the front porch, she put her foot through the good step while missing the bad one. While Danny could get in through the back door, the front hallway was the wider one. Again, Taylor didn't want to risk accidentally wrecking things at home, so Danny went in and got some of her personal belongings after Assault, gentleman that he is, offered some PRT living space meant for things like this.