Speaking about the republic and earlier empire, and quoting Greg Woolf (Food, Poverty, and Patronage: The Significance of the Epigraphy of the Roman Alimentary Schemes in Early Imperial Italy), “neither charity nor state welfare can be shown to have been significant sources of assistance to the poor in the ancient world,” and as u/XenophonTheAthenian noted above, you have to squint very hard at the Augustan grain dole to make it resemble modern welfare systems.
Since you asked about other kinds of welfare, though, let’s at least go over a few other kinds of not‑welfare-except-if-you-squint. Alimentary schemes and public distributions provided by wealthy individuals are noteworthy among other Roman customs and institutions that might have been beneficial to the destitute. However, both are subject to basically the same type of caveats as the grain dole: there is not much evidence that they were specifically aimed at the lowest strata of society, and there is even contrary evidence, namely, that higher-status beneficiaries customarily were given more.
Public feasts and distributions of food or money were often paid for by wealthy members of a community (for instance, a certain town) for the benefit of other members, and in addition to being paid for as one-off gifts or to mark certain celebrations (weddings, holidays, etc.), were frequently set up in a wealthy individual’s will, by establishing a foundation to provide public giveaways in the deceased’s memory on recurring occasions. These foundations often functioned by being initially endowed with a lump sum of capital and then lending it out at interest, over time recycling the initial capital into new loans and using the interest to fund the benefit that had been set up. The benefit to the giver essentially lay in the purchase of prestige, influence, and/or remembrance.
While such distributions of food and money were benefits to the donor’s community, they weren’t welfare and they weren’t even specifically aimed at the poor (which makes sense considering the aim to gain prestige in life or to be remembered after death; neither of these goals required, or was helped by, focusing particularly on the destitute.). The most that can be said of them from a welfare perspective is that a meal or a money gift might have been more important to the very poorest than to others who received it. But these feasts and distributions were occasional events and often offered smaller gifts to the poor than to the influential, whose gratitude was more valuable.
Second, there were various alimenta. An alimentum was a fund set up to benefit (typically) children of a certain community by paying for their subsistence until they come of age. We know of various privately funded alimentary schemes, often aimed at supporting a certain number of children in the donor’s home community. In the reign of Trajan (or possibly his predecessor Nerva) in the early second century A.D., the emperors got in on what had previously been a private and localized game, with numerous large alimenta schemes aimed at children in various communities in Italy.
These alimentary schemes often, though not always, had a very similar financial basis as that of the feast-providing or gift‑providing foundations: an initial endowment of money is loaned out at interest, and the interest income pays for the benefits. Next to the grain dole, these funds for the benefit of children are the Roman institutions that superficially most resemble modern welfare, and they very likely did help some destitute children, but we have the exact same caveats: there is not a lot of evidence that they were specifically for poor children (or even, in the case of the Imperial alimenta, specifically established in poorer communities and regions), and we do have some evidence that the payments were influenced not by how needy the child was but by their status in the eyes of society – such as boys receiving more than girls.
While the grain dole was an enduring institution, these alimentary schemes generally do not seem to have lasted through the Crisis of the Third Century, which is not surprising considering what the crisis meant for the general Roman economy. There were general shocks to the economy and diminished prosperity during the period of military anarchy, which certainly decreased the resources available to set up these foundations and funds, likely reduced the opportunities for profitable loans for established funds, and likely increased the risks of failure to repay the loans at all. Also, for alimenta that relied on paying out interest while preserving and recycling a fixed sum of initial capital, the high inflation of the crisis years would have been even worse than it was for the economy as a whole.
Other sources of financial aid… well, there were other Roman social institutions that may have been relevant to mitigating financial distress – colleges and societies prominent among them, such as the well-known burial societies – but these operated more along the lines of insurance. Members paid regular dues in return for the institution’s covering sudden, major expenses such as funerals. These have their own histories, being both socially and financially important for their members, but they were even less a form of welfare than the institutions discussed above.