I also have another client who wants me to bid shorter term work (they did offer at a rate that's 40% higher) but I told them to hold off for now because I want to get used to the new contract first and make sure I'm comfortable with the cadence.
Key point - I will consistently be raising my rate, but for now I'm making almost as much as my day job and working much less hours, and I need to get out there network, get good, etc... and the rest will come. I'm approaching this in a way that aligns with my own principles. I'm not trying to be a bottom feeder in this market. I just am getting my feet wet with a minimal portfolio and simply happy to be in a position that I can make the transition into something I'd much rather be doing.
This is in Canada though, so I'm not taking the requirement to pay for your own medical insurance into consideration.
i.e., if you make $50k a year, charge $75/hr. If you make $100k/year, charge $150. How much do you think an associate attorney at a law firm that makes $150k per year is billed out at? CPA? Management consultant? All of these professionals are generally billed out at a median rate of nearly $200/hr. And the top end is astronomical, like $800/hr or more.
The hardest part about software consulting/contracting is learning to say a polite "fuck off" to the people not willing to pay your price. Everything else is easy. Making some money is the easy part. Earning respect is the difficult part.
"No less than 5 offers" - CHARGE MORE MONEY. As an engineer it is your responsibility to the craft to increase the prestige and respect of the job.