If you are looking for a tube overdrive sound you won't get it with just a pedal. If you just want an overdriven tone you might be happy with just the pedal.
I would suggest to try it, if it's not enough for you and you have the cash then, go for an all tube head.
I think that "tube overdrive" is overrated when talking of power tube overdrive. For example, when overdriving an SVT (vintage or VR) there are like 2 "Overdrive Stages". First growl is achieved between Volume 11:00 - 11:30. This is the classic SVT tone. Full deep and "rumbling" but yet warm and harmonic sounding it is touch sensitive and not too compressed yet. This is THE trademark of the SVT. And it is indeed diode clipping.

- Caused by a unique feature that the SVT has. It is diodes that are there for protecting the power stage from overdriving.
The power stage is NOT overdriving yet.
Than further on turning the Volume up there come some "brizzly" overtones added to the "SVT-growl". This is the "second stage" . it is when the power stage starts to "knock on the door". The power stage is not fully saturated yet resulting in further increase of volume.
At 2:30 max Volume is reached but not full tube distortion. Still a fullbodied yet not too "unfriendly" sound.
Volume turned all to the right the tone gets really on the angry side. Volume is not louder but seems to be a little louder due to more compression and overtones. Some may call it a little harsh. I love it! THATS finally the typical power tube overdrive, gentlemen.
And then even when Volume is up all the way, there is the active midrange that can even add some more juice.
I play a passive Precision and keep the amp flat except boosting the mids a little and diming the treble. Ch.1 normal - all ultra-something knobs off. I dont like them.
I have a 400 Watt Hiwatt clone, too. This is a PURE 100% all tube amp. PTP hand-soldered (the SVT is NOT an all tube amp when you mind that diodes.

) - This amp stays clean, articulate and a punchy for a long. (no diode clipping there) But when pushed hard the HIWATT gets nasty. Some may like it, but I prefer the "smoother" OD of the SVT.
BTW - The SVT with the fulltone OD Pedal sounds very equal to the sound of the unit when cranked to 3 o clock. So at least for me the PEDAL does it ALL. (at manageable volume)