Fender.
Either a Jazz or a Precision would work fine, although I find Jazzes a bit easier to sit in a mix, and they are a bit easier for most bassists to play. If you got a new MIA (made in America) it'll cost you just about 1K, or you can get a used for around $600 - $650. You can also go CIJ (created in Japan) or MIM (made in Mexico) for less than the American made basses. The Japan builds are usually pretty good, Mexican are said to be hit and miss. So depending on your budget and your preference to US or foreign manufacturing / buying new vs. used you could save a bunch of money OR buy BOTH a Jazz and a Precision. Having a Jazz and a Precision will cover lots of ground, especially in country.
Good luck!