The Via Augusta goes from Cadiz to Seville via Utrera. The Camino Estrecho goes from Algeciras to Cadiz. There's an albergue in Tahiville on the Estrecho, but otherwise nothing. You can read up about them on the
Augusta amigo website.
I walked the Estrecho recently, though I really just made my own way between Algecrias and Cadiz. There were some yellow arrows about, but you need gps. Because you won't meet any other pilgrims and will stay in private accommodation It's far more like a regular hike than a camino, I posted about it here if you'd like to follow my route.
Since you would be staying in regular accommodation on those two routes, the season doesn't make much difference, avoid summer for the best prices and to not get cooked on the Guadalquiver plain. The only point against winter would be that the area around Algeciras will be very wet which could make for a rough first day.
There are a few accounts like Yoah's or the more recent one by Karin309 who walked in winter. It'll be quiet, often cold, and you'll have less daylight for long walking days, but it's quiet doable.