Each deposit hosts a finite amount of gold split between three layers.
The distribution of gold between layers is irregular.
Best predictor is the flavor text associated:

• Layers surveyed as plentiful generally hold a lot of gold if untouched.
  
• Mined layers mean a confirmation of worthwhile gold occurrence and some depletion.
  
• Abandoned depleted layers generally have very slim pickings left to offer.

Excavation costs for each layer are as follows:

• Placer: $15.
  
• Deep: $40.
  
• Hard Rock: $100.

You start out with ×1.0 proficiency for excavating each of the gold layers.
You will eventually encounter three locations that provide proficiency upgrades.
Each location trains for its own layer. You encounter them in a deepening order.
Each provides a single choice of of three upgrades.

• One sets the according proficiency to ×1.25 for $???.
  
• One sets the according proficiency to ×1.50 for $???.
  
• One sets the according proficiency to ×1.75 for $???.

Your proficiency level multiplies your revenue for excavating the according layer.

Unconventional methods are not a permanent ability upgrade as layer proficiencies are.
You go and rent a single use of them once you unlock the locations that provide them.
Once used, you can go rent another use the same way. Rinse and repeat.

Unconventional excavation methods don't create any additional gold.
They move gold from deeper layers to shallower ones, ultimately accumulating in the placer layer.

Their usage offers a potential for some cost saving, given certain conditions.
As a rule of thumb, you save by combining layers.
Moving gold into an already excavated layer requires you to re-excavate it again.
In other words: don't mine any gold until you're happy with the distribution.
This is especially true for hard rock deposits, but more lenient for deep deposits.

Usually moving the gold out of the hard rock layer is the first priority.
All of the unconventional methods aside from Hydro Jets accomplish just that.

The game's official manual, the ingame descriptions and of course the Steam guides make all kinds of mistakes describing the actual effects.

• Hydro Jets from Hydraulic Mining for $25.
  Moves gold from deep layer to placer layer.
  
  
• Drill from Dilapidated mining company for $80.
  It moves gold from hard rock layer to deep layer.
  
  
• Stamp Mill from Mining Partnership for $90.
  I think it moves gold from both deep and hard rock a layer above.
  More explicitly, I assume it moves gold from deep layer to placer layer and then moves gold from hard rock layer to deep layer all in one go.
  In other words, it combines effects given by sequential application of Hydro Jets and then Drill for $15 less.
  
  
• Mercury from Pharmaceuticals shop for $40.
  Haven't tried yet. Supposedly poisons you forever.

Your prospecting license grants you finite permits to engage in excavation.
Each act of conventional excavation exhausts a single permit. That is one for each layer.
Applicable even if you're re-excavating a layer you've already excavated before.
Unconventional methods themselves do not exhaust any permits.
However, their sub-optimal usage may lead to requiring re-excavation.
Despite the flavor text ingame, you shouldn't worry about exhausting the license.
Your license should last you the entire chapter.

We can calculate the investments required for different strategies of exhausting various deposits to compare them.
The calculation includes the cost of renting the unconventional methods used as well as the conventional excavation methods.

• Sticking with conventional methods requires gradual investment of $155.
  
• Using Drill and excavating the remaining layers takes $135.
  
• Using Stamp Mill, Hydro Jets and then placer rinsing takes $130.
  
• Using Drill, Hydro Jets and then placer rinsing takes $120.

The latter strategy seems to be the most economical investment to mine out a full deposit.
The placer rinsing proficiency boosts your rewards with this strategy.
The deep and hard rock proficiencies are not applicable.

• Using Stamp Mill and then excavating the remaining layers takes $145.
  
• Sticking with conventional methods requires gradual investment of $140.
  
• Using Stamp Mill, Hydro Jets and then placer rinsing takes $130.
  
• Using Drill and then excavating the remaining deep layer takes $120.

Notice how you pretty much end up making the same investment for lesser revenue if you premined.
The latter strategy seems to be the most economical investment towards exhausting such a deposit.
The deep excavation proficiency boosts your rewards with this strategy.
If you lack it but possess the placer rinsing proficiency, go with the second best strategy.
The hard rock proficiency is not applicable.

• Using Stamp Mill, Hydro Jets and then placer rinsing takes $130.
  
• Using Stamp Mill and then excavating the remaining deep layer also takes $130.
  
• Using Drill, Hydro Jets and then placer rinsing takes $120.
  
• Using Drill and then excavating the remaining deep layer also takes $120.
  
• Sticking with conventional methods to mine hard rock requires investment of $100.

Notice how for a deposit retaining only the hard rock layer, unconventional methods are economically detrimental.
The latter strategy seems to be the most economical investment towards exhausting such a deposit.
The hard rock excavation proficiency boosts your rewards with this strategy.
If you lack it but possess the deep excavation proficiency, go with the second best strategy.
If you lack that as well but possess the placer rinsing proficiency, go with the third best strategy.

Considering you only receive the revenue of one layer in this case, for an investment almost as expensive as ones for exhausting fuller deposits, dealing with a deposit retaining only the hard rock layer is quite uneconomical. Pre-mining out the two topmost layers is a waste you best avoid.

Whether proficiencies apply to unconventional methods is not ultimately clear but it's probably safe to assume they don't. I'm writing this guide under that assumption.
Which makes the hard rock proficiency nigh-pointless to acquire other than for the purpose of completionism and exhausting a few badly premined deposits, the profit from which isn't likely to cover the proficiency acquisition.

Hiring fellow miners and bringing them along to deposits is another way to boost earnings.
Not to mention they also aid the party in combat as non-essential characters with own guns they bring.
They stick with you to their death.

• Pat Douglas at Town for $200 confers a universal mining bonus of 10%.
  
• Old Ike at Shady joint for $400 confers another 15%.

It is currently unknown whether their bonuses are additive or multiplicative.