As organic matter like broken-down leaf bits and other debris blow or fall into the crevices between stones, they can support weed roots. Therefore, since stone mulches (in this use, the gravel is acting a bit like a mulch) can always support weeds once this happens, the timing of putting-down a new gravel layer might not matter very much with regards to weed prevention. Some weeds germinate in autumn, some in spring, and some can continue germinating all summer, as different species have different life cycles.

Roundup usually uses the chemical glyphosate, which is a systemic herbicide that kills plant roots for hard-to-eradicate perennial weeds. It can work on annual weeds, but usually isn't needed since they die out by winter (or for winter annual weeds, by summer) even if nothing is done to treat them. Glyphosate binds to soil well, meaning it is unlikely to move far from where it was applied and affect other plants or waterways unless there is notable erosion of the soil in a treated area. It also degrades into fairly non-toxic byproducts which will be less hazardous to plants that salt. (Do not use rock salt; other than in trace amounts, sodium is toxic to plants, and if it succeeds in killing weeds, it can kill desirable plant roots as well. It will also be very hard to remove from the soil, which is why farmers on the Eastern shore are having serious problems with saltwater inundation rendering cropland unusable. Additionally, any weeds growing near the shoreline that have natural salt tolerance due to those habitat preferences will not be easily affected by that tactic anyway.)

The simplest way to get rid of the existing weeds, at least for the time being, is to use non-chemical means like heat treatment to kill them. This may require hiring someone with the right equipment (unless you wanted to buy it yourself) and experience using it, and could entail either a flame weeder torch or a steam weeder apparatus. Granted, it would be miserable for the applicator to do this during hot weather, but otherwise, the timing won't matter much, but would be best if done before any current weeds ripen their seeds. Heat won't travel far into the soil, so this won't kill tough perennial weeds, but the loss of foliage to the treatment plus being buried in more stone will probably be enough to weaken most species to the point of not being a problem. Hard-to-kill, tenacious perennial weeds like Canada Thistle should be spot-treated with systemic herbicide before being covered, and treatment just before they flower (so, right around now) tends to be the most effective window of opportunity.

Miri