I suggest seeing how much a local private investigator would charge for a "skip trace" public records search. Most people aren't all that difficult to find. It sounds like you're already taking this step, and perhaps doing a bit more.
Part of the analysis depends on what happened during the child's first two months of life. Assuming this is a Hawaii case (although the provision is not unusual) a presumption of paternity is created if the father "receives the child into his home and openly holds out the child as his natural
child". See Haw. Rev. Stat. Sec. 584-4. If they lived together as a couple raising a child, even if for two months, and the father continued to hold the child out as his natural child after that time, he would appear to meet that statutory requirement. Hawaii has scant case law on the statute.
If a paternity action is filed and the presumed parent (father, mother, or both) appears to be avoiding service, Hawaii law permits the judge to issue an arrest warrant to have that parent brought before the court. See Haw. Rev. Stat. Sec. 584-6(b). Once the parent is brought* before the court they may be required to post a bond. If they subsequently fail to appear, "the trial of, or other proceedings in, the action shall, nevertheless, proceed as though that person were present; and upon the findings of the court it shall make such orders as it deems proper as though that person were in court."
Further, Hawaii does have authority for allowing default judgments of paternity where the mother is given proper notice of the proceedings. See
Doe v. Doe, 98 Hawai'i 144, 151, 44 P.3d 1085, 1092 (2002). (The
Doe case might have had a different outcome had the mother argued and offered some evidence of non-paternity in her attempt to set aside her default.)
How long has she been 'missing'? Depending on the amount of time and her location, the proper venue for the action may be a state we have not yet considered.