With the car analogy, I might think that having driven the route once, I would be more familiar with it on the second run, and that would influence my performance. I can't say whether that is a factor for your pilot study.
In preparing the design, you could do a comparative power analysis of the two possibilities. SPSS Statistics has a power calculator (new in V27). If your measure is a mean, say, you would need to know about what the average and standard deviation are. If it is a proportion, you would need an idea of what the proportion would be. In both cases, you would plug in the sample sizes and the effect sizes or the desired power, and it would tell you the differences between the paired and independent sample designs.
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