Air Carrier First Officer reported a fume event after departure which dissipated shortly thereafter. The fumes returned more intense during descent and the flight diverted to a suitable airport.

Narrative

The Captain and I were operating a Maintenance Ferry from ZZZ-ZZZ1. The Captain was PF (Pilot Flying) and I was PM (Pilot Monitoring). The plane had been sitting in ZZZ for 12 days and undergoing Maintenance due to a [another airliner] hitting the [aircraft's] rudder with its wingtip while taxiing by it. Shortly after takeoff while climbing out of ZZZ; the Captain and I started to smell a strong odor of what we can best describe as oil or the smell of an auto body shop. It smelled like it would when taxiing behind another airplane and that airplanes exhaust gets into our air circulation; however; it was more of an oil like smell and less like fuel. I pushed in the EICAS FULL button to take a look at the oil temperature; oil pressure; and engine vibration indications. Everything seemed normal on the EICAS and we didn't notice any issues with the engine. Above 10;000 ft.; the Captain asked me to take controls and radios while he goes back to the cabin to check if everything is in order. While he was investigating the cabin; I pulled up the ECS Synoptic Page and noticed that the Engine 2 bleed pressure was fluctuating quite rapidly between 0 and roughly 55 PSI while the Engine 1 bleed pressure remained stable at about 35 PSI.When the Captain returned to the flight deck; I brought the fluctuating Engine 2 bleed pressure to his attention. He told me he didn't see anything unusual in the cabin. He took controls and radios while I went to investigate the cabin as well. Before doing so; Engine 2 bleed pressure stabilized to about 35 PSI consistent with Engine 1 bleed pressure. I came to the same conclusion that there was nothing unusual in the cabin. We messaged Dispatch to let them know what we smelled. The Captain and I were discussing possible causes of the odor and initiated a discussion if in hindsight we should've just run the Smoke; Fire; Fumes non normal procedure despite the fact that the odor went away fairly quickly after we noticed it. We both agreed that fumes can negatively affect us and/or incapacitate us without us realizing even if we thought the smell was gone. On the other hand we discussed the implications of prematurely [requesting priority handling]. We came to the conclusion that in the future; it is always best to immediately don the oxygen masks and run the procedure to preserve our health and safety of flight as opposed to the alternative. We discussed how the Smoke; Fire; Fumes procedure doesn't provide guidance on what qualifies as fumes and whether how long it lasts or whether or not it dissipates actually matters when making the decision to run the procedure. We both felt this was a strong oily odor that neither of us had experienced in our careers and in our time flying the aircraft.During an established stable power; stable speed descent into ZZZ1; the smell came back. However; this time it was significantly stronger. We both immediately executed memory items: donned our oxygen masks and established communication. We ran the Smoke; Fire; Fumes QRC and QRH; [we requested priority handling]; and diverted to ZZZ2. I requested ZZZ2 Approach to have the airport roll the [AARF] trucks to inspect our plane after we landed. We landed safely on Runway XXC in ZZZ2 while still wearing our oxygen masks. We had the [airport support] vehicles inspect our airplane and they said everything was normal. At this point we removed our oxygen but kept them handy in case we needed them again quickly. We taxied the airplane to [company] Maintenance Hangar where we spoke with maintenance; [ARFF]; as well as with our company about the incident.I think we should clarify the procedures regarding fumes and strongly encourage crews to immediately don oxygen before attempting to diagnose gene situation and identify the source of the odor and what it is even if the smell goes away quickly as toxic fumes can be harmful and lead to incapacitation quickly and without crews realizing. I think our crews should error on theside of caution to avoid a potential more dire situation leading to adverse effects on health and life.