Air Carrier Captain reported a fume event during climb which dissipated in a matter of minutes. The fumes returned during descent and the flight diverted to a suitable airport for maintenance support.

Narrative

The flight began as a ferry flight on an aircraft that needed finishing rudder repairs in ZZZ. It had been sitting in ZZZ1 for about 10 days before we flew it. The incident was not related to the rudder or control system in any manner. The flight was uneventful from startup through takeoff. On departure climb out we smelt a fume smell similar to that of exhaust smell taxiing behind a 737 with a metal/oil smell similar to a mechanics shop. It was mild at first; becoming worse and then better again all within a matter of minutes. By 10;000 ft. and a few minutes into the flight the smell was mostly gone; though we were unsure if it was because it was gone or if we'd just gotten used to it. We informed dispatch and asked if the COMAT (Company Material) we had in the fwd cargo contained any oil or liquids that could have spilled and recirculated. The FO (First Officer) and I also took turns walking through the aircraft feeling the floor and sniffing around for any evidence of a worsening or threatening condition. From then on through cruise and beginning our descent were all uneventful.We took the time in cruise to debrief what had occurred and discuss what we'd do if it returned. Our consensus was that our checklist doesn't give leeway into; 'how bad is the smell' or 'what is the smell'. Rather it simply states if we have fumes of any kind we don oxygen; QRC which leads to QRH declare and land at nearest suitable. Our plan was after this was simple; if it comes back we go to ZZZ2 or ZZZ; whichever is closer. Advised ATC on descent somewhere in the mid 20s. Prior to this we'd been in a stable (speed/power/rate) descent for 7 or 8 thousand ft. (FL340 - FL20 something). Seemingly out of nowhere the smell returned and was significantly more potent than the first time around. Per our debrief we did our memory items; ran the QRC; and the FO ran the QRH and advised ATC with intentions to go to ZZZ2 (50NM off our right wing) while I executed the divert; set up the box; notified dispatch and briefed the visual approach. We stopped off the runway upon landing to have the trucks inspect the aircraft prior to heading to our maintenance hangar. All remaining or unmentioned phases were uneventful. My final thoughts revolve are just what smell we had. Multiple members of company asked if it was a 'dirty sock smell'. I would not at all characterize what we experienced as anything like that. And coming from a long time hockey player I'm certainly familiar with such atrocious smells! My background also puts me with limited shop experience and a knowledge of the smell of burnt rubber; trans; hydraulic and oil fluids. This was closer to the smell of a 737 exhaust when taxing too close behind them than any other smell I can think of. If I had to liken it to any oil I'd say it was closest to the smell of fresh trans fluid that you'd put in a trans case. It was pungent; but no sock I've come across compares. One of the discussions I had with Management after this situation was in regards to the fumes QRC and this 'dirty sock smell'. Long story short I was informed that this was most likely not an issue and we could have continued. This was absolutely not said in a way that implied what we did as being wrong in anyway. It was more of a 'just so you know to add to your toolbox' type conversation. I was also informed of the potential intent on removing fumes from the QRC and initially I agreed. Heck we just had it on departure and didn't react appropriately. According to our manuals the reaction is simple; 'smell something?' If yes QRC land. We didn't do that. However; after thinking about this topic extensively since then and reviewing some reading material supplied to me by the union I believe what we have now is the best option. Furthermore; I think we should take it more seriously. Consider this; 10 times out of 10 if you smell something and follow the steps we have I am confident you will be safe on the ground to talk about your situation.On the other hand; if you instead decide to figure out what your smelling; how bad the smell is; or if it's a threat; the outcome could quickly become catastrophic if the odor is a serious threat. I'm also confident that there are plenty of dangerous odors that I have never smelled before and some that might not seem dangerous at first discovery. I'm happy with the results of this event. I'm here telling you about it. I'd urge you to think carefully before allowing people to decide which smells are dangerous and which aren't. The safest route is; without any doubt; to some your mask and get the plane to the pavement. A last note; with now near 3;000 hours in this aircraft and 100's if not thousands of takeoffs and landings I have never once smelled anything like this onboard this aircraft or any other I've flown in as a passenger or operator.