In principle the list seems to be fairly open-ended.  A common word not otherwise discussed here is βάση (με βάση τα δεδομένα,  με βάση τη δική μου πείρα)  Others include θέμα (με θέμα Χ) and γνώμονα.  Can any of these be used with a following να?  Another set of words which can be used with με but ?not with έχω seems to be those denoting working relationships:

 με συνεργάτη τον Έλληνα γιατρό Νικόλαο Τρίτο
έβλεπε και το αυτοκίνητο με οδηγό τον Κυριμόπουλο να απομακρύνεταιι
ρεσιτάλ με σολίστ τον Χ (example from Mackridge (1985): The Modern Greek Language. OUP).

In such cases, again there are problems of translation, in the sense that the Greek construction cannot be translated literally.  Other phrases may be possible such as in the first case, in collaboration with; and in the second case, the car, driven by X; the car, with X at the wheel, etc.