Can I do a VFR climb-out on IFR flight plan?
KTUS doesn't have a VCOA, but I can think of a couple of alternatives (other people may have more). First, file a composite (VFR to IFR) flight plan. See the AIM section 5-1-7:


  Flight plans which specify VFR operation for one portion of a flight,
  and IFR for another portion, will be accepted by the FSS at the point
  of departure. If VFR flight is conducted for the first portion of the
  flight, pilots should report their departure time to the FSS with whom
  the VFR/IFR flight plan was filed; and, subsequently, close the VFR
  portion and request ATC clearance from the FSS nearest the point at
  which change from VFR to IFR is proposed.


Second, depart VFR and ask for a pop-up IFR clearance when you need it. But this isn't ideal if you know that you'll be in IMC (or in the flight levels) at some point and you want to make sure your clearance isn't delayed.

Third, depart IFR and just ask ATC for whatever handling and routing you want. If weather conditions are good and there isn't much traffic, there's a good chance they'll let you do what you want. Considering the terrain, though, I wouldn't do it unless you're familiar with the area. And of course there's a chance that ATC won't allow it anyway.

Finally, although this doesn't answer the question you could just fly the ODP. I've never been to Tucson but the airport elevation is 2643' and right now the temperature is 32C and the altimeter setting is 30.10, that means the density altitude is about 5000'. And since you're departing to the east (more or less) it looks like you'll need to get to at least 11000' (IFR altitude) because of terrain, so you have to climb 8000' with limited space to do it and with reduced performance because of the high DA.

I don't know what you fly, but climb performance in those conditions isn't great in many light piston aircraft - at least if they're normally aspirated - and a climb in the hold seems like a good way to get up there safely. Of course, if you know the area well and your aircraft is up to it, then things would look different.