Low copy numbers of plasmids
I am sure you have done it; just a note to others: do a PCR screen to
ascertain that the plasmid is there.
For minipreps you generally do not need a starter culture. 
If you have screened your clones and now just want to amplify and keep the plasmid for cloning purposes then go for a maxiprep with 200-400ml cultures. 
Do not overgrow the cells, your extraction becomes difficult and you would get even worse yields. Just set up a huge culture and pool the cells before extraction. Always seed and harvest cells that are in exponential growth phase.
If you need to screen the plasmid for clones or something then do a colony PCR, keep the colony inoculated in a small culture (make a temporary glycerol stock if your screening is taking time). If screen is positive then incoulate the starter culture for maxiprep/gigaprep culture with this stock.