Somehow today I had a bit more luck in addressing my own question. It seems that many of the online blood testing services do offer tests on a disaggregated basis. Their more popular products are bundled tests that include some of the typical overall blood tests for an annual physical. But, several of those enterprises also offer unbundled tests of more esoteric yet very important tests such as for C-Reactive protein and Fibrinogen. The latter is a very good check to see how granular can you go in terms of specific independent blood tests you can order from such services. Today, I found two companies that passed that check. One is Accesa Labs that operates through Quest Diagnostics. The other one is Health Tests Direct (HTD). HTD is actually very interesting because they operate through both Quest Diagnostics and LabCorp. As a result, HTD allows you to do some comparative shopping in terms of test prices between those two market leaders.
Although the above answer is reasonably inclusive, it does not preclude that others may uncover specific companies that could be equal if not superior to the two I mentioned above.
This is a short update as of December 2017. As we know Theranos has run into all sorts of legal and regulatory troubles since I first wrote this answer. They are now I think very much out of this business. After considering many online lab testing providers I recently chose Personalabs.com. They allowed you to select your blood test on a disaggregated basis as well as any other online competitors. What differentiated them is that their website and their blood test selections were so much clearer than competitors. They operated through Quest Diagnostics and LabCorp just like the other better services. Interestingly enough, I got my lab test results through Personalabs.com within 5 days. Yet, over 10 days later, I still could not get the same from Quest Diagnostics (QD). I am still interested in getting the results from QD just to check that their interpretation of the results (healthy acceptable range of values) are the same as for Personalabs.