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Although I am not going to say much about this issue since I have not read much about it, since I am a Spaniard just wanted to tell you that there is even an older civilization than Tartessos in southern Spain which is Turdetanians. They seem to be related.
And also saying that most people in "Spain" spoke a basque like language: iberian (most people agree that basque was extremely related with iberian). And some theories say that in many places of Europe and previously to the arrival of indoeuropeans, the natives spoke an "iberian" like language.
And there was also a intermedial language called celtiberian, which is related to celtic (it even has PIE Consonants, and iberian: https://en.wikipedia.org/wiki/Celtiberian_language
This is the celtiberian language:
https://en.wikipedia.org/wiki/Botorrita_plaque
llengües paleohispŕniques
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