Not without a Load Calculation
I agree with crip659, you should presume that you can't add another 50A load to this panel. Based on the number of breakers, and particularly, 240V breakers - this panel is likely already near or past limits.
The way to resolve this is to do a NEC Article 220 Load Calculation on the service. You must use the NEC procedure there described, or a municipal worksheet derived from same. Don't even bother trying to contrive some fake way to do a Load Calculation that by wild coincidence works in your favor - everyone tries that.
Here is a typical city-supplied Load calculation worksheet, which is based on NEC Article 220. Don't bother with commerical websites as many of them put up fake clickbait articles that "make stuff up that sounds cool" (understandably since the NEC calculation is pretty dense). Any genuine Load Calc starts with "# of square feet x 3".
https://www.cityofsacramento.org/-/media/Corporate/Files/CDD/Building/Forms/CDD-0213_Electrical-Load-Calculation-Worksheet.pdf?la=en
If your Load Calculation checks out with the hot tub added (which I really doubt) then you can just go ahead and merge circuits onto tandem breakers. Cutler Hammer "CH" is an absolutely first-rate panel, one of the finest industrial panels made, and you need not worry about its age.
"The Load Calculation gave me a hard no"
I'm pretty much expecting this. Don't give up yet, though.
Technology Connections just dropped a video last week about this exact problem, so I'll go ahead and link that here. But the upshot is that you can use energy management systems to exclude loads from the Load Calculation. This technology is only starting to arrive on the market, which is why the $7000 SPAN panel is the only consumer-ready offering. But both Square D and Eaton are spinning up smart-breaker retrofit schemes for their panels. Your Eaton CH will lag a couple years behind BR (the breakers are smaller and it's hard to fit the tech inside) but you'll get it.
But it can be a lot simpler than that.
At its simplest, if two loads cannot be on at the same time, only one counts for the Load Calculation. So this could be as easy as an interlock between two 240V breakers so they can't be on at once - the Eaton CHML interlock does that nicely. E.G. Dryer, or hot tub. Moving breakers around isn't a big deal.
If you need to interlock the hot tub with two appliances, then you can use Eaton CHMLs to interlock the breaker with the one above and below.
Other things are possible - for instance, an older A/C system only runs when a 24-volt signal comes from the thermostat. That signal could also be used to interrupt power to the hot tub, so when the A/C is running, the hot tub is not.
For EVs we get a special treat - actual 21st century tech. (well, 1980s tech really, but nevermind that). EV charging can be taken off the Load Calculation entirely by using a Current Transformer sensor on the supply wires, to automatically slow down EV charging when the house would otherwise be overloaded. This is implemented by Britain's "Myenergi Zappi" and by the USA's Emporia Vue home energy monitor paired with the Emporia EVSE - and that's one of the most affordable EVSEs.
Many ways to solve it but hard to speculate without knowing more about your panel.
A 400A service upgrade could be done as well - this would actually preserve your CH panel - no need to mess with that - and add another 200A panel. You might as well get another CH, they're fantastic panels.