If rhinoplasty is the most difficult facial plastic surgery procedure, then elongating the over-resected nose is the most challenging rhinoplasty. Unfortunately, I think that your case bears this out and although further surgery can result in improvement, it will be very challenging and you will need to seek out a highly experienced revision rhinoplasty surgeon. The decision to proceed with another surgery will not be any easy one and should not be considered until 1 year has passed in order to allow full tissue healing.
I agree with your assessment that the tip is overly up-rotated and that the columella hangs down, both of which result in excessive nostril show. In addition, it appears that the dorsum has been over-resected resulting in a low nasion point, a scooped out or concave dorsal profile and a shortened nose.
Elongating the over-resected nose will require that attention be paid to the 3 layers of the nose, all of which may be contributing to the small, short appearance:
Internal mucosal lining - potentially scarred and shrunken.
Cartilage framework - usually depleted and weakened.
Outer skin cover - invariably will have some degree of scarring with a shrink-wrap effect.
Successful enlargement may require that maneuvers be performed to address one or more of the 3 factors above, depending on their contribution to the problem. The principal causes of failure in attempted elongation of the nose during revision rhinoplasty are a) underestimating the amount of cartilage grafting necessary and b) underestimating the effects of an extensively scarred skin cover which can overcome even the best efforts to restore the cartilage framework in the long term.
In your case, at 3 months following surgery, my guess is that inadequate cartilage grafting was performed because the skin envelope has not yet had the time to shrink the nasal framework. Following a 1 year waiting period, if the nose has not continued to get substantially smaller, then the existing skin envelope may still allow for nasal enlargement. I would recommend harvesting rib cartilage to provide material for the following grafts: a) dorsal onlay graft to reverse the scooped appearance, b) extended spreader grafts to lengthen the supratip region, c) septal extension graft to push the anterior septal angle ineriorly, d) nasal tip onlay graft and possibly e) alar rim grafts to decrease the amount of nostril show.