I'm not sure what choroba said in response to this question, and these comments are in the context of general Perl OO practice and not Moo, but ...

... "Does $self->new make sense?" ...

I sometimes see constructor code that looks something like

sub new {

    my $class_or_objref = shift;

    my $ref_class = ref $class_or_objref;  # true if an object referen
+ce

    my $object_reference;

    if ($ref_class) {
        # build object reference from existing object attributes
        # and supplied @_ arguments.
        }
    else {
        # build object reference for class from supplied @_ arguments.
        }

    return $object_reference;

    }
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To me, this factors naturally as

sub new {

    my $classname_or_objref = shift;

    return ref $classname_or_objref ? $classname_or_objref->new_clone 
+  (@_)
                                    : $classname_or_objref->new_pristi
+ne(@_)
                                    ;

    }
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(Update: From this point on, @_ | no, changed my mind, will use ... (yada) for this notation represents a general argument LIST, possibly empty.)
So every class/package may potentially need both a
    $object_reference->new_clone(...)
method that builds and returns a new object reference based on the attributes of an existing object and any additional arguments supplied, and a traditional
    ClassName->new_pristine(...)
method that builds and returns a new object reference based solely on a class name bareword/string and any arguments. It's immediately obvious that
    ClassName->new_pristine(...)
looks a lot like the very familiar
    ClassName->new(...)
usage, and
    $object_reference->new_clone(...)
can very intuitively be represented simply as
    $object_reference->clone(...)

IMHO, there is no value added by trying to combine object clone creation with "classic" object construction. The calls
    my $obj = ClassName->new(...);
and
    my $obj = $existing_object_reference->clone(...);
should do exactly what you expect them to do and no more. A  ClassName->new constructor definition can often be very short, sweet and easy to understand; don't clutter it up with unnecessary stuff. If no  clone() constructor is needed for a class (and it usually isn't), don't define one. So, in answer to the question,  $self->new can make sense, but usually doesn't, and certainly not with that method name.

... does Perl have something similar to new XYZ; which will search for the nearest new() all the way up the inheritance tree of XYZ?

IIRC, any  ClassName->method() or  $objref->method() call will search for method() first in the specified class (either explicitly named or embedded in the object reference) and then all the way up the @ISA tree of the specified class under the control of the mro (available with Perl version 5.10+) currently selected, and then into the UNIVERSAL class. (And, of course, new has the same status as any other method name. It is used as a constructor name by convention; any other — well documented! — name could be used.)