You can extend it as you have in mind, but the reality is a long unsupported fended will whip around like crazy and may even get itself snagged on the wheel.
Also be aware that water doesn't leave a rear wheel radially, but tangentially. So to do you any material good, you'd have to extend the fender almost all down to the 3 o'clock position. You also would want to extend forward down to the chainstays so water isn't blasted at the bottob bracket.
IME, folks have two choices. They can either go bare, and take whatever nature throws at them, or they can go for full fenders with a flap at the back of the front one. Anything in between will be of marginal benefit, if any.
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