Since you are concerned about costs, I would try to nudge you off novices' preconceptions about copper vs aluminum wire. Novices tend to "stay with what they know" (having worked with copper in 15/20A branch circuits), and tend to believe all the scary stories about aluminum wiring in 15/20A branch circuits. This extrapolates into only ever looking at copper for large feeder, and never even thinking about aluminum feeder.
Actually, aluminum was never a problem with large feeder wires. Heck, the lugs are usually aluminum (since aluminum lugs are a "universal donor" that plays well with both copper and aluminum wire). So insisting on copper "to avoid dissimilar metal problems" actually has the reverse effect lol.
The normal approach for large feeder is aluminum. The power company also uses aluminum pretty much exclusively for distribution, and service entrance wiring is typically aluminum as well.
However aluminum is -2 numerical sizes larger, and will often require physically larger conduit. That appears to be the case here. However the savings on wire will plenty more than pay for the larger conduit.
Code requires that if wire is #6 or smaller, the neutral wire must be natively white color. If wire is #4 or larger, you are allowed to use electrical tape, sleeving or paint to re-mark a black wire as a neutral.
This gives an advantage to #4 aluminum over #6 copper, since you only need to buy one color of wire.
There's actually no such thing as 60A feeder. In fact since you are in conduit using THHN, your choices are 50A, or 65A breakered at up to 70A.
Your #6 copper wire, since you are using THHN in conduit, is actually good for 65A of actual ampacity. Noting this means 65A of 240V... or two poles of 120V at 65A each... nominally 130A of 120V loads.
You can only provision 65A/240V of loads, however, since 65A breakers are not made, you "round up" to the next available size breaker, 70A.
If you want that same ampacity (65A) in aluminum, #4 aluminum will suffice.
If you only need 50A, #6 aluminum will do, but you'll need to take care to get a white neutral wire.
Power is not provisioned at breaker trip capacity. It is perfectly normal to "oversubscribe" a panel, i.e. the individual breakers total to much more than the service or feeder capacity. Look at your own main panel, for instance! That's because not all loads are used at maximum at once.
The upshot is that a 65A feed can support a lot more stuff than you might think. So get a much larger subpanel. Extra breaker spaces are dirt cheap when you're buying the subpanel... chintzing out on the panel and thinking "Oh, I'll just change to a bigger panel later if it comes up" is a whole lot of work.
There is nothing wrong with a 100A or even 200A subpanel on a 50A or 65A feed.
Assemble conduit, then pull
Note that with conduit, you must assemble the entire conduit run and have it complete and finished... before you start pulling any wires into it. This means you must assemble the conduit in a fashion that is pullable. While many conduit novices are aghast at this requirement, I assure you, it's much easier to work with overall. I wouldn't do it the other way even if it was legal!