Unfortunately, drag force is proportional to the square of velocity... So the slingshot (or railgun or whatever) maybe gives you a nice acceleration at the start, but a LOT of that initial force will bleed off as drag early in the flight in the thick thick low atmosphere.

(Consider the difference in difficulty of riding a bike at 15mph vs 16mph and 25mph vs 26mph... 25->26 requires a hell of a lot more effort, all due to drag.)

Or you could just launch from a 12 mile tall tower: https://en.wikipedia.org/wiki/ThothX_Tower

Or a Lofstrom loop: https://en.wikipedia.org/wiki/Launch_loop

Or you could fly a high-altitude airplane into a skyhook: https://en.wikipedia.org/wiki/Skyhook_%28structure%29

There's also using chemical rockets for the first stage, then switching to nuclear for the second stage once clear enough of the ground to minimize fallout. Radiation pollution wouldn't be pleasant, but future advances could potentially minimize that also.

NERVA XE was damn near ready to fly during the Apollo era for Mars missions. Would've made a hell of 2nd stage engine.

https://en.wikipedia.org/wiki/NERVA

Or a comically large refueling hose hooked up until it hits orbit. I guess in that case you'd probably be fighting gravity even more trying to push fuel up the hose.

Ground laser propulsion has been considered. https://en.m.wikipedia.org/wiki/Laser_propulsion