Correct me if I am wrong, I understand that you have plasmids (with high copy number) in E. coli and want expressed high yield protein. So, important thing is how did you design your vector. If you didn 't used an overexpression vector so here is your answer. You need to change promoter site (something like B- actin) for higher and constant expression.
Howver I will add that protein expression is also a function of your insertion cassette and procedures for expressoon:
Some proteins are naturally more toxic and/or more difficult for bacterial vectors to express
The former results in the E coli strain remaining in lag rather than entering exponential or log phase for a protracted period of time and consequently the culture doesn’t grow to a very high density resulting in low protein yield
The latter is partly a function of host strain: pay attention to the type of Ecoli you use and select strains that have been engineered to be compatible with more efficient production of protein e.g. BL21 DE3 or Rosetta
In addition more efficient protein expression tends to result from an initial heat shock followed by expression of protein at suboptimal temp (18-22C) for long periods once you have grown to an initial optimal density at 37C
Finally akways use fresh transformants or at the very least do not pick colonies from agar plates that are more than 1 week old
For certain proteins initial growth of bacterial transformants on skeleton media supplemented with maltose can also help
When I return to work on Monday I can send you my old SOP incorporating some of these suggestions
Besides vector design, you need to consider expression conditions for your target proteins. Expression host strains, media composition, pre- and post-induction temperature and concentration of inducers are some of the critical parameters.