I don't think that the sound change t→ð/V_V is common enough that we can make claims about what is the normal development of the change. I also think that the question of pre-existing contrasts matters. For example, in a number of languages in and around the Niger Delta (Efik, for example), final voiceless stops lenite to [β r γ] between vowels: but /β r γ/ are not underlying phonemes in these languages. Moreover, there is no contrast between voiced and voiceless obstruents finally (underlyingly or on the surface), although the languages do otherwise contrast voicing in obstruents. In this case, since the outcome for the coronal consonant is [r] and not [ð], you may wish to exclude such examples. Numic languages (Shoshoni, Paiute, Comanche) tend to have intervocalic lenition, and the output is usually or always a voiced spirant, and this results in ð or r in Shoshoni and r in Southern Paiute. Again, this is an allophonic process (possibly neutralizing in light of modern loans), so there is no contrasting /θ, d, ð/ that earlier t merges with. One possibility is that t→ð does not arise when there is a contrasting /d,θ,ð/.
We can rule out such a strong statement, since there are languages that contrast voicing and spirantize voiced and voiceless stops. Florentine Italian is reported to have lenition where p t k become voiceless fricatives or frictionless approximants (and b d g likewise becomes voiced fricatives / approximants), but these are non-neutralizing (i.e. there is no underlying φ θ β ð). Berber also lenites stops, t→θ, d→ð. In both cases, the output is an allophone of a stop. The situation with Liverpool English on this point is unclear to me.
Trimming out intermediate processes, there are three plausible paths for getting this change. (1) t→θ,θ→ð; (2) t→d,d→ð; (3) t→ð (one step). There is an account of such lenitions where the sound change is simply shortening of the intervocalic consonant, which results in a "quick fricative" plus passive voicing (see Kirchner Effort-based approach to consonant lenition). You might rule out (1) and (2) in a particular instance if the change did not carry along the hypothesized intermediate output, if it existed as a phoneme. For instance, if you have Ursprache t,θ and t→ð but θ did not also become θ, we could rule out (1) in that language. It may not be an accident that t→ð doesn't arise in languages with d,θ,ð. The shortening theory depends on there being no "need" to distinguish t̆ from d,ð,θ.
