access/modify an object inside of a function | javascript

It sounds like you're trying to modify the behavior of constructor at runtime. You've said:

I would like to modify the code of constructor using dot notation (if its possible in this case), e.g. I would like to add more properties to my constructor. If the code was like this:

var constructor = function(name, surname, town){
    var person = { Name: name, Surname: species, Town: town };
    return person;
};

...then I guess I could modify my constructor like this: {constructor.person.phoneNumber = 554457; but in my original code constructor has no var inside so I cant taget obj

As @deceze pointed out, no, you couldn't do that even if constructor had a variable in it; that variable is entirely private to the function, it's not exposed as a property of constructor.

The only way¹ to modify constructor's behavior at runtime is to wrap it in a new function, like this:

(function() {
    var original = constructor;
    constructor = function() {
        var obj = original.apply(this, arguments);
        obj.phoneNumber = 554457;
        return obj;
    };
})();

That calls the original version with all of the arguments it receives (and the same this), then adds the additional property, and returns it.

¹ Okay, technically, you could decompile it with toString (on browsers that support it; support is now required as of ES5, so it's pretty good), use text manipulation to change its contents, then use new Function to turn it back into a function and assign the result to constructor, but A) It's a really bad idea, and B) The function would lose its context, and so if it relied on closing over any variables that aren't globals, it would stop working (hence [A]).