You basically have it right.

Is there some way to access methods in MyPackage::Module::ThatFeature, short of fully qualifying them as: MyPackage::Module::ThatFeature->still_another_method()?

To explicitly call a method in a superclass, fully qualify the method name with its package:

# $obj is an object which is in a subclass of MyPackage::Module::ThatF
+eature

$obj->MyPackage::Module::ThatFeature::method_name(...);
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If you don't know the name of the superclass, you can use the SUPER pseudo-class:

$obj->SUPER::method_name(...)
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Have a look at perldoc perlbot for more good stuff on perl objects.