How does an aircraft designer increase dynamic stability?
Damping is produced by drag and by large induced speeds at the tail surfaces from a given disturbance. This can be caused by long lever arms of these surfaces or by high air density.
More on the topic can be found here:

What is aerodynamic damping?
Will control surfaces on a plane be less efficient at a higher altitude?. The lower density at high altitude reduces the vertical or sideways speed component caused by a rotation relative to flight speed, thus reducing dynamic stability.
Another trick is to use a conventional tail instead of a V-tail, as discussed here.
What also helps is to shift the eigenfrequencies of the plane away from the response delay of the pilot in order to avoid PIOs. An unusual example would be the Wright Flyer.


These planes typically pull out of a dive on their own, but then climb excessively and stall …

This is the classic long period mode in longitudinal stability. Since rotation rates are low, pitch damping also is low and the most important damping contribution is from drag. A low L/D reduces the tendency to overshoot, a high trim speed reduces the tendency to stall (and shifts the motion to higher speeds with lower L/D). Reducing static stability will make the period longer such that it becomes easier for the pilot to react. However, lower stability will make the pitch response more sensitive which increases the risk of too large control inputs.