SOV languages use suffixes all the time, and they often mark both dependent and head. The Turkish translation of your sentence is:
Delikanlının yaşlı köpeği kara kediyi kovaladı
Young_man-GEN old dog-NOM.POSS3SG black cat-ACC hunt-3SG.PRET
So possession is both marked on the head with a genitive suffix and a possessive marker on the dependent. And languages with default SOV order like Latin can have head marking only, with suffixes, plus genitives and adjectives following their dependents:
Canis senex iuvenis felem nigram persecutus est.
dog-NOM old-M.NOM.SG young_man-GEN cat-ACC black-F.SG.ACC hunt-PERF.PARTC.M.NOM.SG be-3SG.PRES