First, It's not good for you nor for the whole sector to do free works. But I understand that at first is hard to take money from someone who trusts you.
I think that the problem is the questions you are asking and how you are asking them. ThisThese questions are made almost in argot, you are using a very straight forwardstraightforward and professional language.
My approach would be more human. If you have the oportunityopportunity to go to the company to talk withto your client. Look at how they work, how is everything organized, maybe even you will see other things you can improve such as billboards, cards or catalogs and you can suggest to expandexpanding your contract.
But if you can't go there, just call him. Talk to him as a counselor, this way you are selling your own brand. Ask him what they do (I think you've already talked about it), ask him how is the company doing (probably will tell you about problems that you can identify with design or marketing problems, even they tell you about competence) why he thought about requesting your help, what he thought about improving (you've already diddone this for sure) ask him if you like any company style (probably he should have pointed other company logos right now).
The part where you ask about competence in my country at least would feel gross. You have to research the market. Which companies are the best, which ones are the direct competence, what's the difference between that company and your clientsclient's company, how to make that differences matter or not. All that is your work as a designer.
Have in mind that unless you are working for a really big company that has complex inner boureaucracybureaucracy and needs to controllcontrol every investment, you usually won't receive all thisthese answers in a straight forwardstraightforward manner.
As a designer, you won't be only designing. You will be a "friend" of that company. Most of the people would come to you without really knowing their needs, even if they act likeas they know. You have to act as "visual communication therapist" to hear their problems and analyze what they need.