You have to "break in" new tires to get your mileage back. Usually takes a few thousand miles.
If the tires are LRR (low running resistance) you will usually get better mileage with them.
Inflating to higher pressures will also improve mileage, but I wouldn't go over the "max. inflation pressure" on the sidewall. Also, higher pressures can make the tires more susceptible to damage from road hazards. Tires with higher max. inflation pressures are usually better quality and stronger than those with lower max. inflation pressures. For example my Nokian WRs are rated around 54 PSI.
But your assumption is correct. Usually worn tires will give better mileage than "good" tires. Until you kill yourself.
