10K under 40 minutes is blazing, especially for a big dude. You are the man hippie.
Long slow distance would be nice, but I would be careful to ramp it up nice and slowly, not putting on too many miles too quickly. Even if they are slow miles, the time and distance will not be kind to your shin splints. When I start a program after a long hiatus, I do a similar program as you, jogging a half mile and walking about a quarter mile. after a couple of weeks I feel like I am able to slowly ramp it up so I am jogging a mile or two continuous and increase by 10-20% per week after that.
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