The inflation damage has already been done with the "infrastructure" package and other spending. It's a (then somewhat, now entirely) supply side shock, which is the kind of recession one can't spend their way out of.
And so, oddly enough, the blow is right for once, just not in any positive way:
Because the correct response to a supply shock is different than a demand shock.
A very loose talk about how this works:
For why you can't spend your way out of a supply shock problem, it's useful to loo at why government spending does help (at least short term, not arguing with Austrian Econ right now) during demand recessions.
A demand recession is largely just in everybody's head: "Oh, we're in a recession, so money's tight, so I better not spend." Obviously, this train of thought makes logical sense, but it also creates a self fulfilling prophecy because if consumers don't spend, then there isn't enough demand, so companies lower production, so there aren't enough jobs, making money tight, etc.
But government can basically step in and spend oodles of cash, which jumpstarts the system of consumption back to production capacity, which means that the production now needs people to work, which means consumer money isn't tight, and they can consume instead. Now government can leave again (usually it doesn't, but lets ignore that for now).
But a supply shock is 'real'. For whatever reason, companies across the board now cannot produce enough (Gas, war, disease, stupid government regulations, all can cause this). And the usual bad stuff from this occurs. But the difference is if the government increases demand by buying stuff, that doesn't solve the underlying problem. So just as much stuff is produced, just at a higher price. This is inflation.
We actually have both, since the Fed hasn';t tightened and is still propping up the markets with bond purchases, and then we have the very rear supply chain issues, great resignation and now the situation with Russia.
The recession we got when covid and the lockdowns started was, IMO both a demand and a supply side one.
People were scared to spend or lacked liquidity, so moneyprinter go burr.
The disruption to actual production and shipping and the panic buying made things even worse.
Also, money printer go brrr is far from the only way to increase the money supply, artificially low interest rates disincentivise saving, stimulate excessive risk taking and purcahses and create a huge asset bubble, then when people see their purchasing power getting eroded and their neighbors getting rich they join in and further inflate the bubble.
Soros called that type of behavior a virtuous circle, and that was IMO the only decent bit of advice in his whole book.
This, IMHO all happened in the 1960s -1970s, when first the US government spent massive amounts of money on wars, social programs and literal moonshots.
Then, the Arab oil embargo acted as icing on the cake.
I think we are seeing the same situation as the one in the late 60s/early 70s.
Money supply expansion inflation is IMHO something that echoes through the whole system over and over, it does not double the prices right away, but moves through the system and increases demand for the inflated currency as it does.
First you give one set of people the money, they might spend it immediately, or they might do it more gradually, then it goes in the economy, increasing salary demands, first the people with the lowest wages will have to demand more, then those above, and so on and so forth, with IMO the effect working through the whole economy a few times if everything else is stable.
Of course a boom might force a bust depending on timing, money ammount, and where the bulk of the money ends up, but it is not a spigot or a switch.