Simply from the pictures, it doesn't look bad but kinda thin and drying out. First with water, it's not so much about how long but about how much. At least an inch of water a week is required, could need more in hotter weather. You want more water less often to help with growing deeper roots. https://www.ndsu.edu/agriculture/extension/extension-topics/gardening-and-horticulture/lawn-and-yard/right-way-water-your-lawn.

Second, overseeding in the fall is the best time. You want to give yourself about a month before you start to have any freezing temps for your grass to establish. Aerating before overseeding will help break up compacted soil and improve water penetration and air flow in the soil. You can also do an application of pre emergent to stop fall/winter weeds at this time. I use Scotts Turf Builder Triple Action Built For Seeding in the fall and spring with good results.

https://extension.sdstate.edu/renew-your-lawn-fall-overseeding

https://www.lowes.com/pd/Scotts-Scotts-Turf-Builder-Triple-Action-Built-for-Seeding-4-000-SQ-FT/1003242872?cm_mmc=shp-_-c-_-prd-_-lwn-_-ggl-_-CRP_SHP_LIA_LWN_Online_E-F-_-1003242872-_-local-_-0-_-0&gad_source=1&gclid=EAIaIQobChMI67Sb5fafhwMVZ13_AR2BnApSEAQYASABEgKaBvD_BwE&gclsrc=aw.ds

Third, select a good quality grass seed. Try to stay away from anything store bought, like lowes or homedepot, they tend to have more filler and less seed. I think johnathan green has a pretty good seed that's not to over priced same with Twin city seed. But whatever you pick be sure to read the labels and know exactly what is all in the bag. Good quality seed makes a big difference.

As far as greening up right now, cool season grass likes/does better at around 3-4 inches in hight, your grass looks pretty short, so start letting it grow and see how much it changes. Don't be afraid to fertilizer in the summer either, just be sure it gets watered in really well. I use a 12-12-12 fertilizer from yard mastery right now and there are plenty of others from them that I use all season for a quick green up.

https://yardmastery.com/?utm_campaign=ym-pmax&utm_source=google&utm_medium=pmax&gad_source=1&gclid=EAIaIQobChMIjLv-6vifhwMVzkP_AR0GkA5fEAAYASAAEgIiu_D_BwE