How can I improve the overhang angles my printer can successfully print?
The two most important things you can do are:


Provide adequate cooling to solidify the plastic quickly
Minimize layer height


Cooling is really obvious. You need the plastic to solidify before it has a chance to sag. PLA in particular has to shed a lot of heat before it is fully solid. A fan and air guide setup using a "squirrel-cage" radial blower around the nozzle is optimal. A little 30mm or 40mm axial fan will not provide optimal performance. 

Low layer height when slicing is less obvious, but is extremely effective. When you use thinner layers, two things happen:


There is less melted plastic per pass and a higher surface area to volume ratio, so the fresh material cools faster. 
A larger percentage of each strand in the overhang is supported by the previous strand. If you do 0.2mm thick by 0.4mm wide, half of each strand is unsupported. But if you do 0.1mm thick by 0.4mm wide, only a quarter of each strand is unsupported. 


When you combine these two effects, it is possible to exceed 70 degree overhangs with good surface quality. 

Another lesser factor is printing shells/perimeters inside-out rather than outside-in. This helps anchor the outermost strand a little better as the overhang is built. This is pretty minor though.