Engine Won't Crank Due To Starter on Geo Prizm

Noticed battery was dead. Replace that and nothing happens when I start. Replace the starter and turn key - nothing happens again with not even a sound coming from the entire engine. I thought I would at least hear the starter solenoid energies. Out of ideas. Any suggestions. Thanks

Car does not like to start in cool damp weather the last few years after about 260-275K. Once warmed up and/or dry no problem. Starter clicks but does not turn - sometimes just hesitates a second sometimes several tries of 5-10 sec holding key to start position. strong batter helps but did not solve. Previously checked out but it always worked if in garage.
Gets worse every winter. Finally having starter replaced at 290000.
Had been trying to avoid too many more repairs after a major one earlier this year. Need to keep car drivable for at least another year.

Just replaced the old starter. New one works great.

Just had car checked over by mechanic before short road trip with my elderly mother. Everything was great. Stop at grocer for supplies and won't start. Attempted to jump 2x, no success. Replaced the starter and runs good as new. Love my Geo !

Had a new battery installed two yrs ago. For about the last six months, my car didn't want to start at times and I would have to jump start it. The garage came and said it's the starter. They want to replace the starter now. Please note, the battery has a 75-month warranty, but garage swears there's no problem with the battery.