Usually weekend rates are higher than weekday rates so if you are able to take some time during the week to move it is recommended. If the moving company is out of budget then I would consider alternatives such as hiring a van with a trailer as it usually comes out cheaper. The only downside is you would need to source your own people to do the lifting.
The best way to find a good moving company is by looking research on the labor for packing in Los Angeles before you decide to hire them. By doing research on the companies, you will know what their rates are, what services they offer, and how long it will take them to complete the move. You should also make sure that they are insured and have a license.