You're not making it easy asking for both a lower bail-out gear _and_ closer-spaced high gears. That's less easy than either/or, and hoping to do this without changing derailleurs and a whole bunch of other stuff is a tall order. You also haven't hinted at how low a low gear you think you need/want, so all we can do is talk about what we use, and spitball. It's hard to be specific without actual numbers. You will need to know the actual tooth count of the freewheel cogs.

Spacing your high gears closer doesn't impact derailleurs, but can you find the cogs/ratio you want? Like John E said, a corncob pretty much means single-tooth jumps all the way through, so a 6spd corncob would be 13-18. 13-23 is only a corncob if you're talking 11 cogs. Maybe you can find the right cogs for your freewheel to go 13-17+24, corncob plus a bailout. Maybe not. Maybe going from 6spd to 7spd will give you better choices.

Lower gearing is likely to impact ders, both to shift bigger cogs/smaller rings and to handle the extra chain. Going triple means extra chainring/bolts, longer bb at a minimum. Maybe new crankset, maybe new bb. Depending on how small an inner ring you want, possibly new fder. And likely new rder as well, if you want a really low low with a lot of chain to wrap.

Many possibilities. If your triple inner can go down to 24t, you can change your 42 to something bigger and get closer high gears. Go from 6spd to 7spd for closer high gears.

You'll want most of the high gear work on the two bigger rings, and use the small ring for the hills. But you have to pay attention to the gear progression. How many duplicate gears on 13-14-15-16-18-24 (for ex) and 42/52? 44/52? You've got 5 cogs from 13-18 for high gears, and 2 rings for 10 high gears, but if 3 or 4 are identical, or close to it, you've only got 6 or 7. And the shifting necessary to avoid the dupes might be a pain. Crossover makes easier shifting, but wider gear spaces; half-step maintains tighter progression, but shifting is considerably more complex.

This is one of the reasons, like it or not, that compact doubles with 9-11 cogs simplify gearing life. No dupes when you're only using the big ring for faster/flatter riding, dumping to the small ring for climbing. No double shifting to maintain a steady progression, easier A/B big/small shifting in the front, less muss, less fuss.