Try this: start by saying "with the sauce"... practice a few times until that phrase is smooth, then begin to suppress the vowel sound from "the", and keep diminishing the enunciation of "the" until it is completely imperceptible, not even the small "bump" between the first and third words. Now it should be easier to transition from the 'th' of 'with' to the 's' of 'sauce'.

Another way to do it: prounounce "this" slowly and clearly, and gradually drop the voicing of the 'i' ( shorten the vowel sound each time ), eventually resulting in "thssss"; now you have taught your brain to slide from 'th' to 's' more smoothly. Finally, try "with sauce" and see how it goes. :-)

In "General American" the pronunciation varies depending on the letter that follows:

In front of consonants, it usually sounds like wɪθ: "go with (wɪθ) Nancy"

with the possible exception of "L", where it usually sounds closer to wɪð: "go with (wɪð) Linda"

In front of vowels, it's usually wɪð: "go with (wɪð) Adam"

I can't speak for any of the various regional accents or dialects, but where I live the regional speech patterns tend to emphasize consonants, so it frequently comes out as wɪð when speaking slowly. But if they're talking normal to fast, they still tend to revert to the above rules.

The phonemic, i.e. prototypical, pronunciation of the word with in British Received Pronunciation is /wɪð/, with a voiced /ð/, as in the.

However, in contexts like "with people", with may be pronounced with a voiceless [θ], as in think, even in RP. This is because the word people starts with a voiceless consonant, /p/. Unless followed by a voiced sound such as a vowel, the voiced obstruent phonemes of English /b, d, ɡ, dʒ, v, ð, z, ʒ/ at the end of syllables are partially or wholly voiceless, which are then distinguished from the voiceless phonemes /p, t, k, tʃ, f, θ, s, ʃ/ only by when the preceding vowel ends. (See this blog post and the answers to The pronunciation of ending "s" and Are "whores" and "horse" homophones?.)

So if you want to speak RP, it wouldn't be incorrect to use a voiceless [θ] in with before words which start with a voiceless consonant, like people. Strictly speaking, since it ends phonemically with /ð/, the vowel /ɪ/ in with must be slightly longer than that in e.g. myth, which ends in a voiceless /θ/, but since with is a function word, which is most often unstressed and pronounced very quickly anyway, the distinction in vowels is most likely irrelevant in this particular case.

Meanwhile, many American and Scottish speakers pronounce with consistently with a voiceless /θ/, even when followed by a voiced sound. From Longman Pronunciation Dictionary by John Wells (source):