To solve the problem, consider the barriers.
* Financing. This can be solved at local level if cities set incentives for developers to build the type of housing that is locally necessary. However, it would be better fixed at state or local level by streamlining regulations, requiring banks to assist, and/or establishing government programs to boost such developments. If for-profit businesses aren't interested, then it is the government's responsibility to ensure that suitable housing is available to citizens.
* Zoning. Anything humans have made, humans can change. While many people want to keep their neighborhood the way it is, not everyone does. There are plenty of folks who would like changes. So, look for neighborhoods with a lot of people interested in raising the density. Change the zoning to allow that. This will be much easier -- and less destructive -- than forcing zone changes in areas of high resistance. It's desirable to have different types of neighborhoods for different tastes.
* Objections over size, shape, character, etc. of new construction. Aim to make modest, incremental changes that fit well with the current neighborhood. Frex, many duplexes, even some triplexes or fourplexes, look pretty much the same as single-family houses. They can also be made by subdividing big houses. Auxiliary living units can be added above a garage or attached to a house with minimal impact. Adding tiny houses or new freestanding units is more likely to create friction. However, it's much more feasible to take a large empty space and fill it with houses sharing a central courtyard. The houses themselves won't look all that different, and this kind of infill changes one block rather than a whole neighborhood. Ideally, the larger corner lots on each block should hold larger dwellings, whether big houses for big families, a duplex with offset entrances, a small apartment building with 2-12 units, etc. Don't build ugly buildings. Often the older ones look nicer than modern crap. You can't blame people for resisting eyesores. Make it good or butt out.
* Objections over crowding and resources. The natural outgrowth of more housing is more people, so there's no getting around this one. Try to focus on areas where people want more housing, because there are some. Use compensatory joys to make up for the added hassle. For example, a neighborhood getting an infill project might also gain a community center or have an undeveloped field upgraded to a park.
* Objections over traffic and parking. One option in near-grid neighborhoods is to complete the grid by cutting new roads. While it is not ethical to force people out of their homes to do this, many people nowadays aren't as attached to place as they used to be, so they might well be enticed by an offer of cash or simply trading their old house for one newly built. If you only need to convince 1-3 homeowners to clear the space for each connection, it's a good bet.
Another excellent solution is one I haven't seen proposed: housing designed for carless households, something that a growing number of people wish to become. (That's odd, because this type of housing is common around any campus, as many students don't own a car yet.) After all, people who don't own cars will add little or nothing to car traffic. Imagine a small apartment building that has, instead of ground-floor parking, a locked bicycle garage with a bike wash, repair station, and shower room. You could probably fit a laundry room and rec room down there too. Without needing a parking lot, it has more yardspace for a community garden. Or if you're retrofitting an existing apartment, you could turn the parking lot into a skatepark for residents.
For best results, you want to pack amenities closely to create a walkable and bikeable neighborhood. So look for the little cluster of shops and services that most neighborhoods have somewhere, often at the edges. Put the higher-density carless buildings there, right next to a source of jobs and goods. Rowhouses or townhouses are ideal for live-work space. The retail or office on the ground floor makes a more walkable neighborhood. These make a great fit right next to the extant shopping area. Just make sure to connect them with a nice sidewalk. Important point: don't run that alongside the arterial street bordering the neighborhood. People don't want to walk or bike there because it's noisy and smelly even if a solid barrier provides protection from zooming traffic. Done right, the neighborhood becomes a miniature of the city itself: a core of denser housing around the service sector flanked by less-dense housing farther out.
There's even a solution for parking/congestion in larger apartment buildings, already in play if overlooked in this context. Turn the apartment into a mini-village. If it includes one or more amenity floors with recreation, dining, shopping, a clinic, a salon, etc. and one or more business floors for employment topped by residential floors, then that greatly reduces the need for people to travel elsewhere. They might choose to, but fewer of them will be obligated to drive every day for necessities. If the building additionally owns a shuttlebus and a small fleet of cars that residents can use, this will further reduce the need for private vehicles. The available parking spaces could then be rented by those who need them, and carless residents wouldn't have to pay for parking they don't use.
On a related note, large employers should provide housing nearby. If you're building a factory, build some apartments so the new workers have a place to live that isn't onerously far away. Include shuttle service and they don't need a car just to drive to work. Include a bike garage and locker room, then they can bike or walk to work. There are still a few places with this kind of setup. Our Wal-Mart is right next to a cluster of mid-size apartment buildings, filled mostly with students and employees. While the layout doesn't make for a nice walk or bike ride, it is feasible and people still do it. So another way to cut traffic and congestion would simply be to look for near-good situations like this and improve them, thus increasing their usage and decreasing car useage.
Worth mentioning, Terramagne-America handles things very differently.
One key difference is that towns are responsible for meeting the needs of their citizens. They are obligated to provide things like suitable housing, or else they lose access to things like federal project considerations. So if there's a glut of high-end housing but not enough middle or low-end housing, the city would not approve proposals for any more high-end housing, only the kind people actually need. Any developer wanting to work there would have to propose useful plans, not just try to line their own pockets.
Another is that because the town itself often backs necessary projects to get them done, or acquires buildings to meet important needs, towns often hold property that they can use to solve problems. It is much easier and less stressful to distribute 100 homeless people across various houses than to put them all in the same building that nobody wants nearby. In this way it's possible to avoid temporary homelessness from becoming chronic, as can happen after house fires or the like. Similarly if they want to encourage small business development, they just make a building available cheap or free for that purpose. The successful businesses will soon outgrow the space and rent somewhere else, and those that fail won't lose too much.