Hi,
I've spent some time riding my old 80's road bike and I've noticed that it's a stretch for me to reach the brake levers, whether on the hoods or on the drops. From the hoods, I can wrap my last two fingers around the levers to apply some stopping pressure, however if I need to stop fast, I have to slide my hands outwards such that I just have my thumb across the back of the hoods in order to get three or 4 fingers to the lower part of the brake handle to apply the most mechanical advantage. I tinkered a bit with the position of the drop bars, rotating them up helps the position on the hoods, but hurts it on the drops. Even in the optimum position that I've found, if I go to the drops, which I do if I really have to brake, I have to reach my hands upwards a good bit to reach the levers. Initially I thought about replacing the bars with a more tightly rounded set with less 'drop' which you commonly see on current bikes. Then I realized, that while that may help with the drops, it does not really help with the position from the hoods, where I spend more time. Then I thought about cross levers. I've never used those and I do spend a fair amount of time with my hands close to the stem, but I was wondering if they had the same mechanical advantage as the primary levers, ie: could you apply the maximum braking force with cross levers as you can with the road levers?
I've also looked at newer levers and some of the aero levers seem to possibly position your hands closer to the lever when on the hoods, but they still seem to have the same shape as my old levers, so I'm not so sure they would do much for me.

You could get in-line "interrupter" brake levers.
but given your hand size, it might just be worth your while to get a more recent-design brake lever. and they even make them for smaller hands (shorter-reach levers). My recommendation would be the Tektro R341.
These will likely make a big difference compared to 1980s levers.
If you want to also have brake access on the bar tops, you could go for interrupter levers, but you don't need to start there.

Perhaps more comments than answers, but:
Cross/interrupter levers are rated as having the same amount of travel as drop bar levers. But the actual lever is shorter than the average drop bar lever, so you may need to apply more hand force for the same amount of braking. Rarely a problem for people with average hands.

Braking from the hoods isn't really a priority in road bike (parts) design.
Best braking is had from the drops, which is where you're expected to be when you hit the high speeds, which is when you're expected to need the best braking. And high speeds is the priority for (most) road bike design.
If you're doing something that's "contrary" to the spirit of road bike design (like using it for urban, everyday, casual riding), you have to put up with some compromises.

There aren't that many options for those with smaller hands. Some drop bar levers come with wedges that can be installed to move the lever closer to the bar, which really is a so-so fix, as it also reduces the available travel.
The ideal solution would be where you could move the pivot point of the lever closer to the bar. That'd maintain (more of )the arc of travel while improving reach.

the leverage is a bit less MA than your aero brake lever on the end of the cable..

but that means the brake's actuation ratio, moves to the rim with less lever motion.
still can effectively stop, but feel is different..

[my older ones were CNC cut , now the lower cost current ones are smoother shaped levers ...



There are some brake levers that bend closer for smaller hands, to reach from the drops .

dont see the same from Shimano.. there are rubber wedges to put in their brifters
so they dont open so far to reach.. .

currently, there are short ramp bars too so brake lever hood is closer to you. given same stem length..
+ of course different stems..

a good bike shop can help you dial in the fit and order the parts

You can build hand strength with exercise, if you want to go that route instead.