I'd second the notion of flying to some California destination and renting a car to get to Visalia

Major Airports:
LAX (200 miles)
SFO (240 miles)
While getting to tickets to either airport or car-rentals there will be easy, the trip involves long travel through dense urban areas, complex freeway travel, and at least 3.5 hours of driving....with no traffic.

Regional airports:
Fresno FYI (55 miles)
Bakersfield BFL (80 miles)
Both have at least a dozen regional connections via LAX, SFO, Las Vegas, & Phoenix; and a scattering of flights to Salt Lake City, Houston, Denver, etc. While they both have multiple rental agencies, I'd reserve a car well in advance...as they are not apt to have a *lot* of last-minute cars. The drive will be easy and straightforward and less than an hour and a half throught the Central Valley on Highway 99...which will give you a view of CA agriculture (if it isn't fogged in by Tule fog).

Local Airports:

Tulare only has a general aviation field, so unless you plan to charter a small plane in Los Angeles and pray for a local car rental, forget it.

Visalia Municipal Airport, is a little bigger, it's mostly a general aviation field, but does have 3 flights a day to/from Las Vegas via America West Express in a Beechcraft 1900. (tiny prop plane) Enterprise Rent-A-Car is the only game in town for car rentals, and it's a small outlet.

Personally, I'd look for a flight from LAX to FYI or BFL on an alliance partner to your RTW fare...