Common chickweed is an annual that will soon die out on its own, so no intervention is needed if you'd rather wait and let the grass fill in afterwards now that it's resuming growth. Its abundance might suggest the soil in those areas is compacted, poorly-drained (or too wet for some reason), or the lawn is being mown too short (this allows weeds to get more light to grow well since the grass can't shade their seedlings enough).

Using herbicide now, even one formulated for chickweed and similar weeds, might be pointless since the plant will soon die out anyway. (It might not absorb the chemical well if it's already shutting down as it goes to seed.) While removing the plants for the sake of reducing the seed presence for next year is valuable, with a plant this low-growing, there really isn't a good way to do that short of physically removing all of them, which of course would negate the need for any herbicide.

If you needed a pre-emergent herbicide treatment to prevent future chickweed seed from germinating, it would have to be applied in late summer or early autumn, since that is when winter annual weeds like this sprout (even though gardeners tend not to notice them until spring, when they've reached full size and are blooming). Some pre-emergent chemicals cannot be used at the same time grass seed is being put down, since it will prevent them from germinating as well. (This depends on the ingredients, so always look through product label information thoroughly since each herbicide is different.)

The best long-term solution, especially if you wish to avoid or reduce herbicide use, is to improve the health and vigor of the lawn. Why it is struggling now is hard to say without more information, but steps to take include having a laboratory soil test performed, core-aerating just prior to seeding, and overseeding each autumn with recommended cultivars. If you haven't been fertilizing regularly, applying a suitable lawn fertilizer (which will depend on soil test results) twice in autumn will help support good growth and establishment so the lawn becomes more resilient and less vulnerable to stress. The pages linked above will provide more detailed information. Areas with conditions not suitable for a lawn to thrive, like areas that are too shady, too wet, or too filled with tree roots, are best changed-over to other plantings like lawn alternatives.

Miri

Thank you so much. 
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You're welcome.
Have a great weekend!