You could approach insane levels of information processing technology.
Each "virus" particle isn't the same virus. The viruses each unique pieces of a biological computer processing system. The infection has multiple tiers.
Tier 1 is biological compatibility. The same way that most dog viruses do nothing for humans, but moreso; your virus has to be capable of infecting the target and delivering a payload.
Tier 2 is hacking their biology. Unlike traditional viruses that are trying to infect, reproduce, and jump to a new host, you have work to do in their body.
Tier 3 is building a new consciousness.
Think of the Von Neumann interstellar probe problem. Here you send a star wisp to a new star. It starts off with next to no resources, and starts bootstrapping by finding a metal-rich asteroid and building a small factory to build more stuff.
Eventually it builds enough infrastructure to decompress and run an actual intelligence in this new solar system.
Your virus infection is a Von Neumann "interbiological" probe. It builds a system to decompress and run the viral intelligence in in the alien biology of the host body.
The viral intelligence payload is expensive to carry. So like a Von Neumann probe, you might just send enough intelligence to build a giant communications dish, and then upload the consciousnesses afterwards.
In the viral case, you'd invade the body, take over its biology, then arrange for a high-bandwidth transfer or copy of consciousness from another host.
Tier 4 is building a collective consciousness.
Humans have high barriers between each organism, but we still have a multiple "collective" information processing beings; from cultures, to religions, to corporations, to families, to polities. A viral consciousness with the ability to stream itself from one body to another might have a long, slow over-consciousness that isn't confined to a given body.
The thought processes of each individual infected would matter much like the state of each neuron in our brain matters. Individual infected could even go rogue, much like cells in our body can go cancerous or otherwise misbehave. We have an immune system to identify and deal with that problem that works exceedingly well.
We also have a reproductive system that attempts to isolate the noise of each lifetime from damaging the next too badly, cleaning up entropy by gestating a new being every 20-40 years; most cancer and other biological system "wear" problems in the previous generations isn't passed onto the next iteration.
Building such a viral consciousness is difficult to imagine using natural selection pressure.
But a ridiculously advanced information processing civilization (like, Matryoshka brain scale or crazier, like ice giants from a heat death universe) could plausibly design such a thing.
For a plot, imagine ice giants, consciousnesses spread over galactic scales, in a universe long past the stars going out, thinking huge slow thoughts as they extract the entropy of fading black holes. They know they are doomed, but they solve for how to send extremely small (compared to their size) fragments of information to a warm universe. Extremely limited, they find a warm intelligence, and build "Von Neumann probes" out of existing viruses. These infect creatures, modify them to receive yet more data. Slowly (in our time frame) they send extremely compressed images of pieces of their consciousness in an attempt to escape the long, slow death of theirs.
Even planetary-scale hive intelligences are not nearly large enough; to transmit and store even a small fragment of the universe-scale intelligence that they are, they need to swallow the universe.
The physics of this universe are going to be relatively alien, so they also need to develop local intelligence (or hijack it) in order to build a higher bandwidth communication device. In this young, hot universe, consuming even a fraction of its entropy (say, converting an entire galaxy into a inter-dimensional radio receiver) would be a wealth far more than they can afford to spare back home.