Bit of a cheaty answer this, but if you count the Gulf of Mexico as the Atlantic, not that much. Even still, with targeted attacks in the right counties, it's not too bad. You'd either need a 16.9% swing or a 28.2% swing, but if you target right you can do it with less nationally.
Flip Yuma County (2.7% swing), Cochise County (11.3% swing), Hidalgo County (3.3% swing) and Luna County (1.9% swing) along the southern border of Arizona and New Mexico. Then you go along the Rio Grande valley in Texas, taking Hudspeth (10.2%), Pecos (11%), Terrell (16.9%) and Kinney (16.8%). This would take you to the Gulf with a 16.9% swing, but crucially if you make Trump do a bit worse amongst Hispanics (specifically Mexicans in that part of the world), then you'd be able to do it with less swing.
If you want to go actually coast to coast, then you probably need a third party spoiler, perhaps a Texas National Party - but let's assume there isn't one. Parts of East Texas are insanely Republican, and there's no clear path any further North. You can at least try to make progress by taking Atascosa (14.5%), Guadalupe (15.9%) and Hays (0.5%), getting you to Austin. Take Williamson (5%), Bell (7.5%), Falls (16.8%), Robertson (17.5%) and Brazos (11.8%). Here you face a problem - some counties here require colossal swings, so in practice you'd just put extra pressure in these counties. Grimes County needs a 25.6% swing, so the people of Navasota, Texas will get bombarded with adverts. This takes you to Walker county, which is an "easy" 12.3% swing, before another difficult county in the form of Houston (not to be confused with Harris which contains the city of Houston - 120 miles to the south of here). Houston County needs a big swing of 25.5% so the very circular city of Crockett, Texas will get an advert bomb too. The next county you cover is the hardest on the entire route -Cherokee County requires a staggering 28.2% swing, but you can target Jacksonville, Texas and hope for the best. You can now leave Texas via Smith (20.5%), Gregg (20.7%) and Harrison (22%) counties. Fortunately these hard counties are all in the same media market except Harrison and Grimes so you could just focus on that one media market based around Tyler-Longview.
You have now got to Caddo Parish, Louisiana, which in fact did vote Democrat. There is then a natural path east, going via the following Louisiana parishes: Red River (5.2%), Bienville (4.5%), Lincoln (9.8%), Ouachita (12.7%) and Morehouse (5.7%). This has a border that is barely half a mile long with the Democratic voting Chicot County in Arkansas. You have now reached the Mississippi. But we can keep going.
In Mississippi, go due east to Atalla County and take it (10% swing), and then Winston County (6.2%). You have now reached the Black Belt of the Deep South, named after the soil of the area, but the legacy of slavery in the area remains, with a majority in most counties descended from slaves who worked those very fields. These African-American voters vote Democrat overwhelmingly and allow you to easily head towards the Atlantic. You do need to flip three counties in Georgia - Peach (1.5%), Twiggs (0.8%) and Wilkinson (5.1%), but then you can go straight to the coast, crossing into South Carolina and going via Allendale, Hampton and Jasper counties.
I might put up a map later showing the counties that would flip under a 28.2% swing nationally, plus a more realistic 12% nationally, with 8% extra in Texas and 8.2% extra in the target counties.
