I think the best way to achieve this is via magic items, since doing so will leave the rest of your resources available for things you can't buy with gold. I also think you should dip a level of Cleric, which uses the same primary stat, so you can make use of the large advantage prepared casters have when it comes to spell slots. You only need one level as many spells on the Cleric list are also on the Inquisitor list and you don't actually need to be able to cast the spells you prepare as a Cleric, just able to keep them ready for Spelldriving. The spells you keep in this way don't technically even have to be on the Cleric list, but using only ones from that list is likely to keep your DM happier. You also don't technically need to be a prepared caster to use the Mage's Lubrication method, but dipping Cleric is likely to make people feel better about it, and also gives you a bunch of spell slots on its own (for the same reasons I advocate multiclassing below as a last resort). It's also suggested in the archetype's fluff.
(at-will) Mage's Lucubration
The most effective (if cheesy) way of doing this is probably to get an at-will wondrous item of Mage's Lucubration, if custom items are allowed. If not, some other unlimited source will suffice.
The method of use is as follows:
Cast any 5th level spell normally, as an Inquisitor (Commune is a good choice, since you should probably take the time to listen to your God every day anyways and it's on both the Cleric and Inquisitor lists).
Use Lucubration to regain the spell as a Cleric.
Sacrifice the spell for Spelldrive.
You can't regain the spell you sacrifice to Spelldrive with Lucubration, but you don't need to because you can regain the same spell multiple times. You just regain your one casting of the spell with Lucubration as necessary. This is nice because it gives you infinite 5th level spelldrive.
This is basically a less good lower level version of Mage's Lucubration that takes 10 minutes to cast. Use it if your GM allows custom magic items and you can't afford Lucubration or they house-rule Lucubration to only be allowed to fill the same slot once or some such nonsense. It would let you replenish your spellslots between combat. If the 'three additional levels of spells' is allowed to stack, this can grant you 9th level cleric spells. As a result the spell levels almost certainly won't be allowed to stack, but you can still use it to be able to cast 2nd and 3rd level spells off your Cleric list that aren't on your Inquisitor list if you want to.
Obviously you should only use the 'Prepare' option; the 'Retain' option is useless.
Pearls of Power level I
You'll need a lot of them, but you can use them pretty much as above. Keep in mind that you lose a spell slot of a given level (by casting it) before you get any spelldrive benefits, so you need to have at least two pearls before you see any extra spell slots. This gives you access to lots of level I slots. You could also combine it with Mnemonic Enhancer to have lots of level II and/or III slots, but Mnemonic Enhancer is pretty good at that itself.
If items aren't allowed, you can either take that Extra Spell Slots feat Molot was talking about or multiclass more. If you are playing a low-level campaign, I'd actually recommend multiclassing. Multiclass casters are stereotypically weak from a long-term optimization perspective, but very effective at low levels as they have tons more spell slots as anyone else and have a Will save that's twice your level can be really cool in the right situations. If you choose to multiclass I'd recommend Inquisitor/Cleric/Warpriest/Shaman so that you can maximise your spell slot bonuses from high Wisdom. With 20 Wisdom (human, point buy) this would give you 12/12 instead of 4/4/3 slots per day at level 4. Note that you will never cast even level 2 spells if you keep it up, which rapidly becomes a problem as spell quality outstrips spell quantity (around level 7 or so). You will also have very weird stats.
As a side note, increasing your casting stat (Wis) is the most important method of having more slots, especially at high levels. You should get every boost to Wis you can, especially if you are multiclassing as that multiplies the bonus directly. Unfortunately stat boosting items only go up to +6 unless custom items are allowed (and even if they are stats are very expensive) and wishes and such are only good for another +5. There are other methods of increasing your Wis by stacking various spell effects or exploiting strange spell interactions, but they are likely to get a book thrown at you.
Explanation Regarding Spells that Prepare Spells
The crux of this answer is that there are spells that let you have more spell slots, but they specify that those spell slots are prepared spells.
If your ability to do this seems strange: The real trick is convincing the DM that in itself merely casting the spell--no matter the means--allows another spell to be "stored in [the Inq13/Clr1's] mind as though prepared in the normal fashion" even though the character A) can't store spells in his mind of that level, and B) can't prepare spells of that level in the normal fashion. That relies heavily on the DM agreeing that spells overrule class features, which is likely a very tough--but not impossible--sell. It is part of specific-overrides-general but it's still a very optimizing interpretation of the rules.